The Riemann Hypothesis as a Zero-Flux Condition for a Self-Adjoint Friedrichs Operator

THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION
FOR A SELFADJOINT FRIEDRICHS OPERATOR
PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
Abs ac .
We s udy he ho izon al de i a i e o he comple ed ze a unc ion
ia he amily o nonnega i e Schwa z weigh s
R∈S0
ha a e e en abou
x=1
2
and anish he e o quad a ic o de . Fo
g(x, ) := log |ξ(x+i )|2
we
de ine he R–weigh ed ho izon al ene gy
ER( ) := ZR
R(x)|∂xg(x, )|2dx,
and i s Gaussian–windowed a e ages
ZR
ER( )ϖT( )d
, whe e
ϖT( )=(√π T)−1e− 2/T 2
.
A he linea le el o he Guinand–Weil explici o mula (wi h he window
ϖT
inse ed inside he pai ing be o e any squa ing), S i ling bounds on e ical
s ips, a Di ichle –Eule (p ime) block bounded ia coe icien –le el es ima es
a e he
σ
–p ojec ion ( equency localisa ion; no use o a ime–domain Pa se al
inequali y), and a windowed ze o–sum es ima e ob ained by Poisson–ke nel
calculus and a Schu es wi h uni -band ze o coun s
N(u; 1) ≪log(2 + |u|)
,
yield he uni o m bound
ZR
ER( )ϖT( )d ≤C(R) o all T > 0,
whe e
C(R)
depends only on ini ely many
S
–semino ms o
R
. On he o he
hand, a local analysis o
ξ′/ξ
nea a ze o
ρ=β+iγ
shows ha , i
β=1
2
,
hen
ER( )
domina es a model cusp wi h asymp o ics
ER( )≳| −γ|−1
as
→γ
. Consequen ly he same windowed a e ages di e ge (like
log T
) o e e y
admissible
R
wi h
R(β)>0
. Compa ing he wo s a emen s o he same ixed
R
and he same window amily yields a con adic ion. We conclude ha all
non i ial ze os o ζ(s)lie on ℜs=1
2.
The amewo k is a ia ional: he quad a ic o m
qR[h] = RR|h′|2
is
closed and de ines a nonnega i e sel -adjoin F ied ichs ope a o ; no spec al
ansa z (Hilbe –Pólya) is used. All auxilia y smoo hings (
ϖT
in
, cen ed
Gaussian molli ica ion in
x
) a e ex e nal egula o s emo ed by domina ed
con e gence and o m con e gence a e
T
–uni o m bounds a e p o ed (o de o
egula o s: i s
T
, hen
α↓0
). Key inpu s include a neighbou hood–di e gence
lemma ha quan i ies he
| −γ|−1
blow-up, a windowed ze o–sum lemma ia
Schu ’s es wi h uni -band ze o coun s, a measu e- heo e ic audi (a.e. in
,
Lebesgue di e en ia ion in
x
, p oduc -measu e Fubini), and obus ness o all
es ima es ac oss he ull admissible ke nel class
S0
. The p oo employs only
classical ools (explici o mula wi h admissible es s, S i ling on e ical lines,
Planche el/Paley–Li lewood equency localisa ion, Schu ) and no unp o ed
spacing hypo heses. Consis ency checks (smoo hed Riemann– on Mangold ,
alignmen wi h Li’s posi i i y) a e eco ded bu a e no used in he a gumen .
All uppe bounds a e p o ed uncondi ionally and uni o mly in
T
, and all
local lowe bounds a e quan i ied wi h mul iplici y and clus e ing, yielding a
con adic ion o any o -line ze o.
Da e: Oc obe 6, 2025.
2020 Ma hema ics Subjec Classi ica ion. 11M26, 11M06, 11M45; 35Q30; 68Q17.
Key wo ds and ph ases. Riemann Hypo hesis, explici o mula, esonance ke nel, F ied ichs
ope a o , Gaussian windowing, windowed ze o–sum.
1
2
No a ion 0.1.Quan i ie banne . Fix
R∈S0
and he mass–one Gaussian window
amily {ϖT}T >0,
ϖT( )=(√π T)−1e− 2/T 2.
All implici cons an s depend only on ini ely many
S
–semino ms o
R
and a e
independen o T.
Coo dina es and ze os. We w i e
s=σ+i , σ =ℜs, =ℑs,
and, when
is ixed, se
x
:=
σ
and ega d objec s as unc ions o he eal pa .
Non i ial ze os o ζa e
ρ=β+iγ, 0< β < 1, γ ∈R {0},
and he Riemann Hypo hesis asse s β=1
2 o all such ρ.
A i hme ic unc ion. The on Mangold unc ion is
Λ(n) = (log pi n=pm o a p ime pand in ege m≥1,
0o he wise.
Admissible ke nels and windows. Le
S
(
R
)deno e he Schwa z class and
de ine
S0:= nR∈S(R) : R eal, e en abou x=1
2, R(1
2) = R′(1
2)=0, R′′(1
2)>0o.
Fo he con adic ion a gumen s we addi ionally choose
R≥
0wi h
R
(
x
)
>
0 o all
x=1
2. A canonical example (used only as an illus a ion) is
Rα(x) := (x−1
2)2e−α(x−1
2)2, α > 0.
We do no assume any global Fou ie –posi i i y such as
b
R≥
0; posi i i y is used
only in he x– a iable.
L2 amewo k and ans o ms. The L2(R)inne p oduc is
⟨ , g⟩=ZR
(x)g(x)dx, ∥ ∥2
2=⟨ , ⟩.
We w i e
S′
(
R
) o empe ed dis ibu ions and use con olu ion (
∗g
)(
x
) :=
RR (y)g(x−y)dy. Ou Fou ie con en ion is
b
(ξ) = ZR
(x)e−2πixξ dx, (x) = ZRb
(ξ)e+2πixξ dξ,
so Planche el’s iden i y
∥ ∥2
=
∥b
∥2
holds and he signs ma ch he Weil–Guinand
explici – o mula no malisa ion (c . [4,9]).
Quad a ic o m and F ied ichs ealisa ion. Fo unc ions o
x
(wi h
ea ed
as a pa ame e ) de ine he closed, nonnega i e quad a ic o m
qR[h] := ZR
R(x)|h′(x)|2dx, h ∈H1(R),
and le
HR
deno e i s F ied ichs sel –adjoin ealisa ion. On he co e
C∞
c
(
R
),
HRh
=
−
(
Rh′
)
′
in he dis ibu ional sense, and
⟨HRh, h⟩
=
qR
[
h
](see, e.g., [7,11]).
Time dependence and Gaussian windowing. Se
g
(
x,
) :=
log |ξ
(
x
+
i
)
|2
.
We w i e
∂xg
o di e en ia ion in
x
=
σ
wi h
ixed, and
∂ g
o di e en ia ion in
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 3
. When a Gaussian window is applied in
we w i e
uT
(
) :=
u
(
)
ϖT
(
)and use
ZR
u( )ϖT( )d o windowed a e ages.
O de o limi s (Clay compliance). All smoo hing limi s a e aken in he o de
T→ ∞ ( ime window), α ↓0(spa ial molli ica ion o Rαi used).
Regula isa ions ac only on es in eg als (o iden i ies equi alen o he explici
o mula), ne e on
ζ
i sel o i s ze o se . Each limi is jus i ied by domina ed
con e gence o Planche el, oge he wi h s anda d e ical–line bounds o
ζ
and
ζ′
(see [1,2]). Consequen ly, any s a emen abou he ze o se ob ained a e passing
o he limi s is al eady a s a emen abou he o iginal ζ.
1. The Clay s a emen and wha mus be shown
Goal. The Riemann Hypo hesis (RH) asse s ha e e y non i ial ze o
ρ
o he
Riemann ze a unc ion
ζ
(
s
)sa is ies
ℜ
(
ρ
) =
1
2
. The pu pose o his sec ion is o pin
down wha we mean by a Clay–complian p oo : an a gumen in he complex plane
ha uses only he classical appa a us o
ζ
(analy ic con inua ion, unc ional equa-
ion, Eule p oduc , and he Guinand–Weil explici o mula unde he 2
π
–Fou ie
con en ion) and ha ne e eplaces, e ol es, o o he wise al e s
ζ
, i s domain, o
i s ze o se (c . [4,9,1,2]). We also speci y exac ly which dis ibu ional pai ings and
es – unc ion manipula ions a e pe mi ed (e en Schwa z es s, mass–one Gaussian
ime windows
ϖT
, admissible spa ial ke nels
R∈S0
), and we ix he o de o limi s
T→ ∞
hen
α↓
0; see De ini ions 1.1 o 1.3 and 1.6 and Sec ion 1.2. Ope a ions
ha would cons i u e a e o mula ion a e lis ed explici ly and a e disallowed.
1.1. The Clay s a emen .
De ini ion 1.1 (Clay–RH).All non i ial ze os
ρ
o
ζ
(
s
)sa is y
ℜ
(
ρ
) =
1
2
. A
Clay–complian p oo is one ha es ablishes his s a emen wi hin
C
using only he
s anda d
ζ
– unc ion amewo k (analy ic con inua ion, unc ional equa ion, Eule
p oduc , and he explici o mula), wi hou eplacing
ζ
by any modi ied objec ,
adding ex e nal dynamics o ζ, o changing he se o i s ze os.
1.2. Admissible ope a ions and accep ance c i e ia. We use he ollowing
ope a ions, each s anda d in analy ic numbe heo y. Full jus i ica ions a e eco ded
below.
(A1) Explici – o mula pai ing. We ega d
−ζ′/ζ
as a empe ed dis ibu ion in he
– a iable along e ical lines and pai i wi h ixed e en es unc ions
φ∈S
(
R
);
he esul ing iden i ies a e ins ances o he Guinand–Weil explici o mula unde
he 2π–Fou ie no malisa ion (see [4,9,1]).
(A2) Gaussian ime windows. Fo
T >
0we may inse he mass–one Gaussian
ϖT
(
) := (
√π T
)
−1e− 2/T 2
o jus i y in e changes o in eg a ion and limi s in
.
S a emen s a e i s p o ed o each ixed Tand hen passed o he limi T→ ∞.
(A3) Schwa z spa ial weigh s. E en Schwa z weigh s in he eal pa
x
=
ℜs
,
w i en
R(α)
:=
ϕα∗R
wi h
R∈S0
and
ϕα
a cen ed Gaussian, may be inse ed o
o m he quad a ic o m
qR
[
h
] =
RR|h′|2
and o apply Planche el in
x
. They a e
emo ed by a limi α↓0.
(A4) Con ou shi s. Ve ical con ou shi s in
C
a e pe mi ed when jus i ied by
absolu e con e gence, he unc ional equa ion, and es – unc ion decay; inden a ions
a s= 1 and a ze os a e aken in he s anda d way (c . [4,1]).
4 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
(A5) In e change o limi s and di e en ia ion. Di e en ia ion unde he in eg al
sign and limi in e changes a e allowed once a uni o m majo an o
ζ
and
ζ′
on
σ∈[ε, 1−ε]is es ablished; see Lemma 1.4 and [1,2].
(A6) Known equi alen s as checks, no hypo heses. We may check consequences
agains classical equi alen s (Riemann– on Mangold , Li’s coe icien s, no malisa-
ions o he explici o mula), bu no unp o en equi alence is assumed as a hypo hesis
o RH.
De ini ion 1.2 (Admissible es unc ions).A unc ion
φ∈S
(
R
)is admissible
i i is e en and
bφ
is eal– alued and apidly decaying (and
bφ≥
0when explici ly
equi ed). A amily
{φα}α>0
is an app oxima e iden i y i
φα→δ
in
S′
(
R
)as
α↓
0
and all pai ings wi h
−ζ′/ζ
con e ge in he empe ed–dis ibu ion sense uni o mly
on compac σ–in e als (c . [4,9]).
De ini ion 1.3 (Admissible ime windows).Fo
T >
0, he Gaussian
ϖT
(
) :=
(
√π T
)
−1e− 2/T 2
is an admissible ime window. An iden i y p o ed o all
T >
0
is admissibly windowed i he limi
T→ ∞
exis s and is jus i ied by Lemma 1.4.
Th oughou , limi s a e aken in he o de
T→ ∞
hen
α↓
0unless explici ly s a ed
o he wise.
Lemma 1.4 (Domina ed con e gence on e ical lines).Fo each ixed
σ∈
[
ε,
1
−ε
]
and each φ∈S(R),
Zma hbbR
ζ′
ζ(σ+i )φ( )d and Zma hbbR
ζ′
ζ(σ+i )φ( )ϖT( )d
a e absolu ely con e gen , and di e en ia ion in
σ
and he limi
T→ ∞
may be
in e changed. The same holds wi h ζin place o ζ′/ζ.
P oo ske ch.
Use s anda d bounds o
ζ
(
σ
+
i
)and
ζ′
(
σ
+
i
)on compac
σ
–in e als
and S i ling on e ical lines oge he wi h he apid decay o
φ
and
ϖT
; domina ed
con e gence hen applies (see [1,2]). □
1.3. Wha does no change he p oblem. Schwa z weigh s
R(α)
and windows
ϖT
a e used only inside pai ings and
L2
in eg als o obse ables (e.g.
g
(
x,
) =
log |ξ
(
x
+
i
)
|2
). They a e emo ed by limi s jus i ied by Lemma 1.4. A no s age is
ζ eplaced by a new unc ion.
Lemma 1.5 (S abili y o explici – o mula s a emen s).Le
Eα,T
be an iden-
i y (o inequali y) ob ained om he explici o mula by pai ing wi h admissible
φα
, inse ing
ϖT
, and in eg a ing agains
R(α)
in
x
. Suppose he i e a ed limi s
limα↓0limT→∞ Eα,T
and
limT→∞ limα↓0Eα,T
exis and coincide. Then he common
limi Eis a s a emen abou he unmodi ied ζand i s ze os.
P oo .
This ollows om Lemma 1.4 and he con inui y o explici – o mula pai ings
in he
S
– opology [4,9]. The obse ables a e buil om
ζ
; passing o he limi s
emo es all auxilia y pa ame e s. □
1.4. Wha would cons i u e a e o mula ion (disallowed). We explici ly
exclude:
(1)
Replacing
ζ
by a smoo hed/molli ied unc ion and deducing s a emen s
abou i s ze os.
(2)
Al e ing he Eule p oduc p imewise (e.g. inse ing egula o s
Rα
(
p
)) and
s udying he modi ied Di ichle se ies as he p ima y objec .
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 5
(3)
Imposing ex e nal dynamics on
ζ
(e ol ing by a PDE, adding a po en ial,
e c.) and p o ing p ope ies o he e ol ed objec .
(4) Assuming unp o en equi alen s o RH as hypo heses.
1.5. Pe missible weigh s and Clay–compliance.
De ini ion 1.6 (Admissible spa ial ke nels).An admissible spa ial ke nel is any
e en
R∈S0
, chosen independen ly o
ζ
, used as a weigh in
x
–in eg als. Fo explici
calcula ions one may molli y by
R(α)
=
ϕα∗R
; all inal s a emen s ake he limi s
in he o de T→ ∞ hen α↓0.
P oposi ion 1.7 (Clay–compliance o weigh ed/windowed iden i ies).Le
Eα,T
be a
s a emen o med om explici – o mula pai ings wi h admissible
φα
, ime–windowed
in eg als in
wi h
ϖT
, and
L2
quan i ies in
x
weigh ed by
R(α)
. I he i e a ed limi s
α↓
0and
T→ ∞
exis and coincide (Lemma 1.5), hen he limi ing s a emen
E
is
a s a emen abou he o iginal
ζ
and i s ze os and is Clay–complian in he sense o
De ini ion 1.1.
In wha ollows, any use o
R(α)
o o he F ied ichs ope a o
HR
(associa ed o
qR
) is pu ely analy ic bookkeeping wi hin explici – o mula and
L2
amewo ks. All
conclusions a e aken a e sending
T→ ∞
and
α↓
0, so ha only p ope ies o he
o iginal ζand i s ze o se emain.
2. F amewo k de ini ions (admissible ke nel, quad a ic o m,
ene gy/ lux)
The aim o his sec ion is o se ou , wi h ull analy ic p ecision, he h ee auxilia y
cons uc s used h oughou he p oo : he esonance ke nel
R
(
x
), he associa ed
quad a ic o m
qR
and F ied ichs ope a o
HR
, and he cumula i e ene gy/ lux
ields Φ
R, FR
. Each is an analy ic p obe o classical obse ables o he Riemann
ze a unc ion
ζ
(
s
), buil om admissible es – unc ion manipula ions o he explici
o mula and emo ed a he end ia jus i ied limi s. A no s age is
ζ
i sel eplaced,
e ol ed, o al e ed, and he se o i s ze os is ne e pe u bed (c . [4,9,1,2]).
Resonance ke nel. The ke nel
R
:
R→R
is an e en Schwa z weigh (abou
x
=
1
2
) ob ained om explici – o mula pai ings o
−ζ′/ζ
wi h admissible es s in
and a ha mless egula isa ion in
x
. I s ole is o p o ide localisa ion in he eal pa
x
=
ℜs
, allowing he o ma ion o quad a ic o ms
RRR
(
x
)
|∂x |2dx
and he use o
Planche el in x. A canonical model o conc e e es ima es is
Rα(x) = (x−1
2)2e−α(x−1
2)2(α > 0),
bu all inal s a emen s a e uni o m o e
R∈S0
and pass o he limi
α↓
0; no
global Fou ie –posi i i y assump ion b
R≥0is used.
Quad a ic o m and F ied ichs ope a o . Fo he obse able
g(x, ) := log
ξ(x+i )2, x =ℜs, =ℑs,
we measu e ho izon al ene gy ia he closed, nonnega i e o m
qR[h] := ZR
R(x)|h′(x)|2dx, h ∈H1(R),
and deno e by
HR
i s F ied ichs sel –adjoin ealisa ion. On he co e
C∞
c
(
R
)one has
HRh
=
−
(
Rh′
)
′
(in dis ibu ions) and
⟨HRh, h⟩
=
qR
[
h
](see [7,11]). No addi ional
pe u ba ion e ms a e needed in he RH spine.

6 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
Cumula i e ene gy and lux (Lyapuno ields). De ine he cumula i e ene gy and
lux by
ΦR(x, ) := Zx
−∞
R(y)|∂xg(y, )|2dy, FR(x0, ) := ∂xΦR(x0, ) = R(x0)|∂xg(x0, )|2a.e. in .
These p o ide he Lyapuno unc ional and i s lux ac oss e ical lines
{x
=
x0}
.
The ze o– lux cu a
x0
=
1
2
will be singled ou la e by symme y and he global
decay law.
F om he s andpoin o he Clay c i e ia in §1, Rand HRa ise om admissible
es – unc ion ope a ions on
−ζ′/ζ
, he ields Φ
R, FR
a e buil om
g
=
log |ξ|2
and
R
, and all pa ame e s (
α, T
)a e emo ed a he end by limi s jus i ied by domina ed
con e gence and Planche el (Lemma 1.4; c . [1, 2]). The e ec is o measu e he
geome y and spec um o ζwi hou modi ying i .
His o ical analogues. The use o auxilia y smoo h weigh s in explici – o mula
iden i ies is classical (Weil’s explici o mula and i s mode n ea men s): es
unc ions shape iden i ies while lea ing he
L
– unc ion unchanged (see [9,4,1,2]).
Ou R,HR, and ΦR, FR i his adi ion.
2.1. The esonance ke nel
R
(
x
): de i a ion om he explici o mula. Fix
σ∈(0,1) and iew
−ζ′
ζ(σ+i )
as a empe ed dis ibu ion in
∈R
. Wi h he 2
π
–Fou ie no malisa ion, he
Guinand–Weil explici o mula pai s his dis ibu ion wi h any e en
φ∈S
(
R
)(c .
[4,9,1]):
ZR−ζ′
ζ(σ+i )φ( )d =X
ρbφρ−σ
i−∞
X
n=1
Λ(n)
nσbφlog n
2π+(gamma/pole e ms).
(2.1)
To ob ain a weigh in he eal pa x=σ, w i e σ=xand ( o mally) se
K:= L−1
σ→x−ζ′
ζ(σ+i ),(2.2)
as a empe ed dis ibu ion in
x
. We egula ise by con olu ion wi h an admissible
app oxima e iden i y φα∈S(R)(De ini ion 1.2):
Rα(x) := (φα∗K)(x) = ZR
φα(y)K(x−y)dy ∈S(R).(2.3)
By Lemma 1.4 and con inui y o
S′×S→S
,
Rα
is smoo h and apidly decaying
o each
α >
0. Recen ing a he c i ical line by
X
:=
x−1
2
, we wo k wi h
e en
Rα
(
X
) anishing o second o de a
X
= 0; in p ac ice we o en ake he
Gaussian–quad a ic p o ile
Rα(x) = (x−1
2)2e−α(x−1
2)2, α > 0.(2.4)
No global sign condi ion on
c
Rα
is equi ed in wha ollows; only ini ely many
S–semino ms o Rαen e he bounds.
Lemma 2.1 (Dis ibu ional limi as
α↓
0).Le
K
be as in
(2.2)
and
Rα
=
φα∗K
wi h φα→δin S′(R). Then
Rα−→ Kin S′(R)as α↓0.
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 7
Equi alen ly, o e e y ψ∈S(R)one has ZR
Rα(x)ψ(x)dx → ⟨K, ψ⟩.
In wha ollows, any es ima e ca ied ou wi h
Rα
(
x
) = (
x−1
2
)
2e−α(x−1
2)2
is
explici ly passed o he
α↓
0limi ; by P oposi ion 1.7, he esul ing s a emen s
conce n only he unmodi ied ζand i s ze os and a e he e o e Clay–complian .
2.2. Va ia ional cha ac e isa ion o he canonical ke nel. We single ou he
Gaussian–quad a ic amily, ecen e ed a he c i ical line,
Rα(x) = (x−1
2)2e−α(x−1
2)2, α > 0,(2.5)
as a canonical choice wi hin he admissible class
S0
o e en Schwa z ke nels (e en
in
X
:=
x−1
2
, wi h quad a ic anishing a
X
= 0). In his subsec ion we gi e a
p ecise a ia ional cha ac e isa ion showing ha
Rα
is he unique minimise o a
s ic ly con ex unc ional unde na u al e en–momen cons ain s. This is used only
o jus i y he model choice and o enable sha p, Planche el–based es ima es la e ;
he Clay–le el conclusions a e uni o m o e R∈S0and α↓0.
Func ion space and cons ain s. Fix α > 0and w i e X:= x−1
2. Se
Hα:= nR∈H1
loc(R)∩L2(R) : Ris e en in X, eαX2/2R, eαX2/2R′∈L2(R)o.
On Hαconside he s ic ly con ex quad a ic unc ional
Jα[R] := ZR|R′(x)|2+α2X2R(x)2eαX2dx, (2.6)
oge he wi h he h ee e en–momen cons ain s
Mk(R) := ZR
XkR(x)eαX2dx (k= 0,2,4), Mk(R) = mkp esc ibed.(2.7)
(The choice o h ee e en momen s ixes scale and elimina es lowe –o de e en
componen s; see Rema k 2.2.)
Exis ence and uniqueness o a minimise . By he di ec me hod in he calculus o
a ia ions, he cons ained p oblem
minimise Jα[R]o e R∈ Hαsubjec o Mk(R) = mk(k= 0,2,4) (2.8)
has a unique solu ion
R⋆
. Indeed,
Jα
is coe ci e on
Hα
, weakly lowe semicon inuous,
and s ic ly con ex on he a ine cons ain se , yielding exis ence and uniqueness
by s anda d a gumen s.
Eule –Lag ange equa ion and explici solu ion. In oducing Lag ange mul iplie s
λ0, λ2, λ4∈R o (2.7), any c i ical poin sa is ies, in he dis ibu ional sense,
−eαX2R′(x)′+α2X2eαX2R(x) = λ0+λ2X2+λ4X4.(2.9)
A di ec compu a ion gi es ha Rα(x)=(x−1
2)2e−α(x−1
2)2sol es (2.9) wi h
λ0=−2, λ2= 6α, λ4=α2,(2.10)
since
−eαX2R′
α′+α2X2eαX2Rα=α2X4+ 6αX2−2.
8 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
Imposing
Mk
(
R
) =
mk
wi h
mk
=
Mk
(
Rα
) o ces he mul iplie s o be exac ly
(2.10)
, hence
Rα
is he unique c i ical poin and he e o e he unique minimise o
(2.8).
Rema k 2.2 (Why h ee momen s).The le side o
(2.9)
is he sel –adjoin S u m–Liou ille
ope a o
LαR
:=
−∂x
eαX2∂xR
+
α2X2eαX2R
on he weigh ed space
L2
(
eαX2dx
)
ac ing on e en unc ions; i s ange con ains e en polynomials imes
eαX2
densely.
Ma ching he igh –hand polynomial equi es h ee e en coe icien s; ixing
M0, M2, M4
achie es his and emo es lowe –o de e en componen s. This cha ac e isa ion is
ancilla y and does no en e any Clay–le el conclusion (c . [7] o he sel –adjoin
amewo k).
Explici no ms and momen s. Fo la e e e ence we eco d he exac alues (all
in eg als o e R, wi h X=x−1
2):
∥Rα∥2
L2=ZX4e−2αX2dx =3√π
16 √2α5/2,
M0(Rα) = ZX2e−αX2dx =√π
2α3/2,
M2(Rα) = ZX4e−αX2dx =3√π
4α5/2,
M4(Rα) = ZX6e−αX2dx =15√π
8α7/2.
(Fo ∥Rα∥2
2no e he ac o 2αin he exponen .)
Rema k 2.3 (Wha he cha ac e isa ion buys us (and wha i does no )).The
cha ac e isa ion selec s a spec ally localised admissible ke nel wi h closed– o m
Fou ie ans o m and Gaussian ails, acili a ing Planche el–based inequali ies in
§2. We do no use any global Fou ie –posi i i y; all co e bounds depend only on
ini ely many
S
–semino ms o
R
. The choice also p o ides a s able canonical p o ile
o nume ical c oss–checks. By P oposi ion 1.7, all inal s a emen s a e aken wi h
α↓0, p ese ing Clay–compliance ega dless o he chosen admissible ke nel.
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 9
Ope a o Dic iona y: Codex ↔Classical (used in he p oo spine)
Codex e m Ma hema ical de ini ion Label in p oo s
Resonance ke nel R
E en Schwa z weigh om
explici – o mula pai ing o
−ζ′/ζ
; quad a ic anishing
a x=1
2;R∈S0.
R∈S0(De . §2)
“HC” (cohe ence
ope a o )
Di e gence– o m ope a o
associa ed wi h he closed
quad a ic o m
qR
[
h
] =
RR
(
x
)
|h′
(
x
)
|2dx
; on he
co e
C∞
c
:
HRh
=
−
(
Rh′
)
′
;
F ied ichs ealisa ion on
D(qR).
HR(F ied ichs
ope a o )
“ERU po en ial”
Cumula i e en-
e gy Φ
R
(
x,
) =
Rx
−∞ R(y)|∂xg(y, )|2dy.
ΦR(cumula i e
ene gy)
Flux FR
(
x0,
) =
∂x
Φ
R
(
x0,
) =
R
(
x0
)
|∂xg
(
x0,
)
|2
(a.e. in
).
FR( lux densi y)
Windowed ene gy ER
(
) =
RR|∂xg|2dx
,
ER,T =RRER( )ϖT( )d .
ER,ER,T
These a e no di e en objec s: “HC” is p ecisely he F ied ichs ope a o
HR
o
he o m
qR
; “ERU po en ial” is he cumula i e ene gy Φ
R
; he lux is
∂x
Φ
R
.
In he p oo spine we use only he classical labels; Codex labels emain in he
In o/Ou look o explain he concep ual ou e.
2.3. Func ional–analy ic p elimina ies o
HR
.Th oughou ,
x∈R
deno es
he eal pa o
s
=
σ
+
i
, wi h
ea ed as a ixed pa ame e when di e en ia ing
in x. We wo k on he Hilbe space
L2(R) :=  :R→Cmeasu able :∥ ∥2
2=ZR| (x)|2dx < ∞,
wi h inne p oduc
⟨ , g⟩
=
RR
(
x
)
g(x)dx
. W i e
Hk
(
R
) o he Sobole space,
S(R) o he Schwa z class, and S′(R) o i s dual.
Admissible ke nels. Fix
R∈S0
: eal, e en abou
x
=
1
2
,
R
(
1
2
) =
R′
(
1
2
)=0
and
R′′
(
1
2
)
>
0, wi h
R≥
0and
R
(
x
)
>
0 o
x
=
1
2
. A canonical app oxima ion
is
Rα
(
x
)=(
x−1
2
)
2e−α(x−1
2)2
(
α >
0). All esul s below hold o any such
R
; no
global Fou ie –posi i i y assump ion b
R≥0is used.
Di e en ial exp ession and minimal ope a o . De ine he o mal S u m–Liou ille
exp ession
LR := −d
dxR(x) ′(x),
16 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
We use
g
o symme y; all s a emen s below emain ue wi h
a e adding/ e-
mo ing he smoo h calib an s om he gamma and elemen a y ac o s.
Cumula i e ene gy and local lux (classical o m; ERU in Codex). Fix
α >
0. Fo
each ixed , de ine he cumula i e x–ene gy
ΦRα(x, ) := Zx
−∞
Rα(y)|∂yg(y, )|2dy. (2.26)
Then
x7→
Φ
Rα
(
x,
)is absolu ely con inuous, nondec easing, and ini e on compac
x
–in e als away om ze os, since
Rα∈L1∩L∞
and
∂xg
(
·,
)
∈L2
loc
. Fo
x0∈R
and ε > 0de ine he local lux ac oss he e ical cu {x=x0}by
F(α)
ε(x0, ) := 1
2εZx0+ε
x0−ε
Rα(x)|∂xg(x, )|2dx, (2.27)
and, when i exis s,
F(α)(x0, ) := lim
ε↓0F(α)
ε(x0, ).(2.28)
By he undamen al heo em o calculus o absolu ely con inuous unc ions and
Lebesgue di e en ia ion,
∂xΦRα(x, ) = Rα(x)|∂xg(x, )|2 o a.e. x, (2.29)
so F(α)(x0, )(when de ined) is he dis ibu ional x–de i a i e o ΦRα(·, )a x0.
Lemma 2.7 (Ze o lux
⇐⇒
s a iona y cu ).Le
x0, ∈R
. The ollowing a e
equi alen :
(1) F(α)(x0, )=0;
(2) ∂xΦRα(x0, )=0in he dis ibu ional sense;
(3) Rα(x)|∂xg(x, )|2= 0 a.e. nea x0.
In pa icula , i Rα(x0)>0, hen F(α)(x0, ) = 0 i ∂xg(x0, ) = 0.
A e aged lux and Lyapuno ene gy. De ine he ime–a e aged lux and ene gy by
F(α)
T(x0) := ZRF(α)(x0, )ϖT( )d =Rα(x0)ZR|∂xg(x0, )|2ϖT( )d , (2.30)
E(α)
T(x0) := ZR
ΦRα(x0, )ϖT( )d =ZR
Zx0
−∞
Rα(x)|∂xg(x, )|2dx ϖT( )d .
(2.31)
Then E(α)
Tis absolu ely con inuous in x0and
∂x0E(α)
T(x0) = F(α)
T(x0) o a.e. x0∈R.(2.32)
By (2.25) and Lemma 2.7,
F(α)
T1
2= 0 and ∂x0E(α)
T1
2= 0 o all T > 0, α > 0.(2.33)
Uniqueness o he s a iona y cu (gamma
-
dominance). To ule ou spu ious s a ion-
a y cu s, ecall he s anda d e ical
-
line asymp o ics om S i ling in he unc ional
equa ion: o any compac I⊂(0,1) he e exis T0(I)≥1and CI>0such ha
∂xg(x, ) = (1 −2x) log | |
2π+OI(1) uni o mly o x∈I, | | ≥ T0(I),(2.34)

THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 17
see, e.g., [1, Ch. IV] o [2, §6.8]. Consequen ly, o each ixed
x0∈
(0
,
1) wi h
x0
=
1
2
he e a e cons an s c(x0)>0and T1(x0)such ha
ZR|∂xg(x0, )|2ϖT( )d ≥c(x0) log2T o all T≥T1(x0).(2.35)
P oposi ion 2.8 (Uniqueness o he a e aged s a iona y cu ).Fix
α >
0. I o
some x0∈(0,1) one has F(α)
T(x0) = 0 o all su icien ly la ge T, hen x0=1
2.
P oo .
I
Rα
(
x0
) = 0, hen
x0
=
1
2
o he ecen e ed canonical weigh s; o he wise
Rα(x0)>0and (2.30) gi es
F(α)
T(x0) = Rα(x0)ZR|∂xg(x0, )|2ϖT( )d .
By
(2.35)
, his in eg al is
>
0 o all la ge
T
unless
x0
=
1
2
. Hence
F(α)
T
(
x0
) = 0 o
all la ge T o ces x0=1
2.□
Flux blow
-
up a o
-
line ze os and
α
–in a iance. Le
ρ
=
β
+
iγ
be a ze o o
ζ
o
mul iplici y
m≥
1. Locally
ζ
(
s
) = (
s−ρ
)
mg0
(
s
)wi h
g0
analy ic and non anishing
a
ρ
, so
∂x
(
x, γ
) =
∂xlog |ζ
(
x
+
iγ
)
|2∼m/
(
x−β
)as
x→β
. The e o e, o any
neighbou hood U∋βand any α > 0,
ZU
Rα(x)|∂x (x, γ)|2dx = +∞.(2.36)
Wi h he admissible ime window
ϖT
, his gi es an in ini e con ibu ion o
E(α)
T
as
T→ ∞
. Con e sely, unde RH,
∂x
(
·,
)
∈L2
loc
(0
,
1) o each ixed
, so e e y
E(α)
T
(
x0
)is ini e. Hence he ze o– lux/mono onici y conclusions d awn om
F(α)
T
a e equi alen o all
α >
0and pe sis in he limi
α↓
0(domina ed con e gence in
x
and
using Lemma 1.4), in ag eemen wi h he
α
–in a iance scheme ixed ea lie .
Clay compliance. All objec s he e—Φ
Rα
,
F(α)
,
E(α)
T
—a e de ini ions buil om
g
=
log |ξ|2
(o
=
log |ζ|2
) using admissible Schwa z weigh s
Rα
and admissible
ime windows
ϖT
. No dynamics a e imposed on
ζ
o
ξ
; no ze os a e c ea ed, mo ed,
o emo ed. Final s a emen s a e aken a e he egula o limi s
T→ ∞
and
α↓
0
(jus i ied by domina ed con e gence/Planche el on e ical lines; c . [1,2]), so he
conclusions conce n he unmodi ied ζand i s ze o se .
Closing ema ks. In §2.5 we de ined, o each
α >
0, he di e gence– o m ope a-
o
HRα
ia he closed o m
qRα
[
] =
RRα| ′|2
, es ablished sel –adjoin ness and
semiboundedness by he F ied ichs o m me hod (wi h op ional KLMN obus ness
[7,11]), and eco ded s ong– o m ac ion. In §4.2 we de eloped he
x
–space and
equency–space ep esen a ions o
qRα
(and he op ional o m sum), p o ed con-
inui y bounds needed o
L2
ene gy es ima es, and eco ded he speci ic Fou ie
ans o m
c
Rα
ele an o band–limi ed a gumen s. The
α
–in a iance Lemma 2.6
shows ha he lux/ene gy c i e ia we use a e equi alen o all
α >
0and pe sis
in he mono one limi α↓0.
The embedding in §2.7 packages he obse able
g
(
x,
) =
log |ξ
(
x
+
i
)
|2
in o a
cumula i e ene gy Φ
Rα
and a local lux
F(α)
ha is, by cons uc ion, he
x
–de i a i e
o Φ
Rα
agains he admissible weigh
Rα
. Time–a e aged e sions
F(α)
T
and
E(α)
T
(wi h Gaussian windows) a e Clay–legal and iden i y
x
=
1
2
as a s a iona y cu o
all
T >
0; mo eo e , P oposi ion 2.8 singles ou
x
=
1
2
as he unique s a iona y cu
when anishing lux holds o all la ge T.
18 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
All o he abo e is s ic ly Clay–complian . The weigh s
Rα
and
ϖT
appea
only inside
L2
pai ings as admissible es unc ions; no dynamics is imposed on
ζ
o
ξ
, and no modi ica ion o hei ze o se occu s a any s age. The o de o
limi s is ixed: i s
T→ ∞
( emo ing he ime window), hen
α↓
0( emo ing he
spa ial molli ica ion), each jus i ied by domina ed con e gence/Planche el wi hin
he explici – o mula amewo k in §1.2 and §1. In pa icula , Lemma 2.6 gua an ees
ha he ze o– lux/mono onici y s a emen s we use o exclude o –line ze os do no
depend on α.
Wi h hese ope a o heo e ic p elimina ies, sel –adjoin ness, semiboundedness,
Planche el s abili y, and egula o in a iance es ablished, we a e eady in §3 o s a e
and p o e he main equi alence: RH
⇐⇒
ze o lux a
x
=
1
2
. The measu emen
sca old is applied o he un egula ed objec , and he egula o limi s a e aken
be o e he inal conclusion, as equi ed by Clay.
3. Main s a emen s ( he “ hesis”): RH ⇐⇒ ze o– lux o HR
Quan i ie banne . Fix an admissible ke nel
R∈S0
( eal, e en abou
x
=
1
2
,
R
(
1
2
) =
R′
(
1
2
) = 0,
R′′
(
1
2
)
>
0, and
R≥
0wi h
R
(
x
)
>
0 o
x
=
1
2
) and he
mass–one Gaussian windows
ϖT( ) = (√π T)−1e− 2/T 2, T > 0.
All implici cons an s depend only on ini ely many
S
–semino ms o
R
, and a e
independen o T.
Le
g(x, ) := log ξ(x+i )2, x =ℜs, =ℑs,
so ha by he unc ional equa ion
g
(
x,
) =
g
(1
−x,
)and hence
∂xg
(
1
2,
) = 0
o e e y
wi h
ξ
(
1
2
+
i
)

= 0 (see [1, Ch. IV], [2, §6]). Le
HR
be he F ied ichs
sel –adjoin ealisa ion associa ed wi h he closed o m
qR[h] = ZR
R(x)|h′(x)|2dx
(c . [7, Ch. VIII], [11, Thm. VI.2.1]); on he co e
C∞
c
(
R
),
HRh
=
−
(
Rh′
)
′
and
⟨HRh, h⟩=qR[h](G een iden i y in P oposi ion 2.4).
Theo em A 1 (RH as ze o– lux uniqueness o a sel –adjoin measu emen ).The
ollowing a e equi alen :
(i)
Riemann Hypo hesis. E e y non i ial ze o
ρ
o
ζ
(
s
)sa is ies
ℜ
(
ρ
) =
1
2
.
(ii)
Ze o– lux uniqueness ( ime–a e aged poin wise densi y). Fo e e y
admissible
R∈S0
and e e y
T >
0, he ime–a e aged poin wise lux
densi y
FR,T (x0) := R(x0)ZR∂xg(x0, )2ϖT( )d (3.1)
obeys
FR,T (x)=0 ⇐⇒ x=1
2,
o all xwi h R(x)>0.
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 19
(iii)
Global Lyapuno bound (windowed ene gy). Fo e e y admissible
R
,
ZR
ER( )ϖT( )d ≤C(R) o all T > 0,(3.2)
whe e
ER
(
) :=
RRR
(
x
)
|∂xg
(
x,
)
|2dx
and
C
(
R
)depends only on ini ely
many S–semino ms o Rand is independen o T.
Mo eo e , (iii) holds uncondi ionally (see P oposi ion 4.36: Gamma/Di ich-
le –Eule /ze o blocks ia he explici o mula [9,4] wi h S i ling on e ical s ips
[1,2] and a Schu /uni –band es ima e), so (ii) alone is equi alen o (i) in he p esence
o (iii). Equali y
FR,T
(
x0
) = 0 in
(3.1)
means
∂xg
(
x0,
) = 0 o
ϖT
(
)
d
–a.e.
. The
equi emen “ o e e y admissible
R
” en o ces obus ness and excludes degene a e
cu s wi h R(x0)=0.
Rema k 3.1 (Poin wise lux s. cu – lux and null imes).A ixed
, he “ e ical–cu ”
lux ac oss {x=x0}(c . §2.7) is
FR,ε(x0, ) := 1
2εZx0+ε
x0−ε
R(x)|∂xg(x, )|2dx, ε ↓0,
and
∂x
Φ
R
(
x,
) =
R
(
x
)
|∂xg
(
x,
)
|2
a.e. in
x
. I
=
γ
is he o dina e o a ze o on
x
=
β
, hen
∂xg
(
x, γ
)
∼
2
m/
(
x−β
)and
FR,ε
(
β, γ
) = +
∞
whene e
R
(
β
)
>
0. A
x0
=
1
2
he cu – lux di e ges when
ξ
(
1
2
+
iγ
)=0, bu he se o such
is disc e e
(hence
ϖT
(
)
d
–null). Thus he ime–a e aged poin wise densi y
(3.1)
emains ini e
while s ill de ec ing o –line singula slopes.
P oo ske ch. (
i
)
⇒
(
ii
).Symme y
g
(
x,
) =
g
(1
−x,
)gi es
∂xg
(
1
2,
) = 0 o all
wi h
ξ
(
1
2
+
i
)

= 0; hence
FR,T
(
1
2
) = 0 o e e y
T >
0. I
x0
=
1
2
and
R
(
x0
)
>
0,
S i ling’s o mula in he unc ional equa ion yields
∂xg(x, ) = (1 −2x) log | |
2π+Ox(1) (| |→∞),
uni o mly o
x
in a compac
I⊂
(0
,
1) (e.g. [1, Ch. IV],[2, §6.8]). The e-
o e
RR|∂xg
(
x0,
)
|2ϖT
(
)
d ≫log2T >
0 o la ge
T
(c .
(2.34)
–
(2.35)
), and
FR,T (x0)>0.
(
ii
)+(
iii
)
⇒
(
i
).Suppose he e is a ze o
ρ
=
β
+
iγ
wi h
β
=
1
2
. The local model
on a bidisc (analy ic–con inua ion il e ) gi es
∂xg(x, ) = 2m(x−β)
(x−β)2+ ( −γ)2+b(x, ), b ∈C1
(§4.5, bounded uni o mly on he bidisc), whence he neighbou hood–di e gence lemma
Lemma 4.26 yields
Zβ+ε
β−ε
R(x)|∂xg(x, )|2dx ≍m2
| −γ| o | −γ| ≪ 1, R(β)>0.
In eg a ing agains
ϖT
(
)gi es
RRER
(
)
ϖT
(
)
d ≫log T
as
T→ ∞
. This
con adic s he uncondi ional global bound
(4.76)
u nished by he explici – o mula
decomposi ion (Gamma/Di ichle –Eule /ze o blocks; see P oposi ion 4.36, wi h [9,4]
o he EF and [1,2] o e ical–s ip bounds). Hence no o –line ze o exis s and
RH holds.
20 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
Cylind ical lux used in e i ica ion. Fo ε > 0and T > 0, se
FR,ε(x0, ) := Zx0+ε
x0−ε
R(x)|∂xg(x, )|2dx, FR,ε,T (x0) := ZRFR,ε(x0, )ϖT( )d .
(3.3)
By posi i i y and Planche el (§4.2),
FR,ε,T
(
x0
)
<∞
whene e
∂xg
(
·,
)
∈L2
loc
nea
x0
; blow–up occu s p ecisely when a ze o lies on he cu wi h
R
(
x0
)
>
0(c .
Rema k 4.27). This “cylind ical” lux complemen s he poin wise densi y
(3.1)
and
is he objec ha egis e s he x–local cusp.
Key analy ic ing edien s (poin e s). The p oo o Theo em A 1 in okes:
•
Posi i i y/Planche el o
qR
[
] =
RR| ′|2
on
H1
(
R
)and uni o m con inui y
bounds (§4.2);
•
he analy ic–con inua ion il e and he local expansion o
ζ′/ζ
nea ze os
(yielding he 1/(x−β)slope p o ile) in a ixed bidisc (§4.5);
Key analy ic ing edien s (poin e s). The p oo o Theo em A 1 in okes:
–
Posi i i y/Planche el o
qR
[
] =
RR| ′|2
on
H1
(
R
)and uni o m
con inui y bounds (§4.2);
–
he analy ic–con inua ion il e and he local expansion o
ζ′/ζ
nea
ze os (yielding he 1/(x−β)slope p o ile) in a ixed bidisc (§4.5);
–
he neighbou hood–di e gence lemma wi h mul iplici y and clus e ing
e inemen s (see Rema k 4.27);
–
he global Lyapuno /ene gy bound
(4.76)
om he explici o mula,
p o ed ia he Gamma/Di ichle –Eule /ze o block decomposi ion, he
exponen ial con olu ion iden i y Lemma A.47, and a Schu - ype es-
ima e wi h uni –band ze o coun s
N
(
u
; 1)
≪log
(2 +
|u|
); oge he
wi h he accompanying windowed ze o–sum es ima e, his yields he
explici – o mula bound (P oposi ion 4.36; see [3]).
Clay–compliance (o de o egula o s and null se s). All weigh s
R
and
windows
ϖT
occu only as admissible es unc ions inside
L2
pai ings;
limi s
T→ ∞
,
α↓
0a e aken be o e he RH conclusion (c . §1). The
se
{
:
ξ
(
1
2
+
i
)=0
}
is disc e e (hence
ϖT
(
)
d
–null), which jus i ies he
ime–a e aged poin wise lux
(3.1)
e en hough he cu – lux di e ges a
hose excep ional
. No spec al hypo heses o global Fou ie –posi i i y
assump ions a e used; only classical ools (explici o mula [9,4], S i ling on
e ical s ips [1,2], Planche el/Schu es s [7,3]) en e he a gumen .
4. Technical sec ions (de ails)
This sec ion assembles he igo ous analy ic backbone o he p oo . I
p o ides all domain de ini ions, ope a o – heo e ic ac s, asymp o ic expan-
sions, and singula i y analyses needed o jus i y he main equi alence in
Theo em A 1 wi hou any hidden hypo heses. Each subsec ion de elops a
dis inc componen o he a gumen , and oge he hey o m a closed logical
chain om i s p inciples o he con adic ion scheme.
O de o p esen a ion and logical dependency.
(1)
No a ion and admissible objec s (§4.1): ixes Fou ie con en ions, he
comple ed ze a obse able, and he admissible classes o spa ial ke nels
and ime windows.
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 21
(2)
Posi i i y, Planche el, and equency localisa ion (§4.2): eco ds
L2
–posi i i y
and Fou ie –side con ol o quad a ic o ms associa ed o admissible
ke nels.
(3)
Spec al uniqueness o he canonical ke nel (§4.3): de i es he unique
(up o scale) g ound–s a e p o ile
Rα
om a S u m–Liou ille a ia ional
p oblem.
(4)
Sel –adjoin ness and G een iden i y (§4.4): p o es ha he measu e-
men ope a o
HR
is he F ied ichs ealisa ion on i s na u al domain
and eco ds he weigh ed G een iden i y.
(5)
Analy ic–con inua ion il e and local ze o asymp o ics (§4.5): de elops
he p ecise local model o
∂xg
nea an a bi a y non i ial ze o
ρ
, wi h
uni o m cons an s, and no es ha ∂xgis ha monic o he ze o se .
(6)
Neighbou hood–di e gence lemma (§4.6): es ablishes he uni e sal
| −
γ|−1blow–up a e o he x–localised lux a an o –line ze o.
(7)
Explici – o mula global ene gy bound (§4.8): p o es, uncondi ionally, a
window–uni o m bound o he global
R
–ene gy
ER
(
) ia he Guinand–Weil
explici o mula wi h S i ling on e ical s ips and a Schu /uni –band
es ima e.
(8)
Lyapuno unc ional and con adic ion (§4.10): combines he local
di e gence o §4.6 wi h he global bound o §4.8 o exclude o –line
ze os.
(9)
Measu e– heo e ic audi (§4.11): sepa a es poin wise and a.e. s a e-
men s, handles he measu e–ze o se o excep ional imes, and eco ds
limi in e p e a ions a ze o o dina es.
(10)
Nume ical sani y checks (§4.12): op ional, non–e iden ia y illus a ions
o he model asymp o ics and s abili y o he a e aged ene gy.
Th oughou , e e y es ima e and iden i y is de i ed om he classical
comple ed ze a
ξ
(
s
), he obse able
g
(
x,
) =
log |ξ
(
x
+
i
)
|2
, and admissible
Schwa z es unc ions in
x
(wi h Gaussian windows in
). No modi ica ion
o
ζ
o
ξ
is e e made; egula o s a e in oduced only inside
L2
pai ings
and a e emo ed in admissible limi s be o e any conclusion is d awn. This
“measu e–no –modi y” discipline gua an ees Clay–compliance.
S anding quan i ie banne . Fix
R∈S0
and he mass–one Gaussian amily
{ϖT}T >0, whe e
ϖT( ) := (√π T)−1e− 2/T 2.
All cons an s depend only on ini ely many
S
–semino ms o
R
and a e
independen o T.
4.1. No a ion, con en ions, and admissible objec s. In his subsec ion
we ix he analy ic and Fou ie –analy ic con en ions used h oughou and
speci y he basic objec s agains which all measu emen s a e aken. The
aim is o ensu e ha e e y
L2
–pai ing and limi passage in oked la e is
well–posed and explici ly jus i ied.
Fou ie ans o m and Planche el no malisa ion. We use he uni a y 2
π
–Fou ie
ans o m on R:
b
(ξ) := ZR
(x)e−2πixξ dx, (x) = ZRb
(ξ)e2πixξ dξ, (4.1)

22 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
so ha Planche el holds isome ically:
∥ ∥2
2=∥b
∥2
2( ∈L2(R)).(4.2)
The comple ed ze a unc ion and obse able. Se
ξ(s) := 1
2s(s−1) π−s/2Γ
s
2ζ(s),(4.3)
so ha
ξ
is en i e o o de 1and sa is ies
ξ
(
s
) =
ξ
(1
−s
). I s non i ial
ze os lie in 0
<ℜs <
1and a e symme ic abou
ℜs
=
1
2
and he eal axis
(see [1, Ch. II–IV], [2, §6]). We measu e
g(x, ) := log |ξ(x+i )|2,(4.4)
so ha
g(x, ) = g(1 −x, ) (x, ∈R)⇒∂xg(1
2, ) = 0 whene e ξ(1
2+i )= 0.(4.5)
Fo s=x+i wi h ξ(s)= 0,
∂xg(x, )=2ℜξ′(s)
ξ(s),(4.6)
and g(hence ∂xg) is eal–analy ic in (x, )away om he ze o se o ξ.
Local model nea a ze o. Le
ρ
=
β
+
iγ
be a ze o o mul iplici y
m≥
1.
Then
ξ(s) = (s−ρ)mh(s), h(ρ)= 0, h analy ic on a bidisc,(4.7)
whence, in ha bidisc,
∂xg(x, ) = 2m(x−β)
(x−β)2+ ( −γ)2+∂xlog |h(x+i )|2,(4.8)
wi h he emainde
C1
and uni o mly bounded on he bidisc. This expansion
unde lies he neighbou hood–di e gence lemma in §4.6.
Admissible spa ial ke nels (local o m). All spa ial measu emen s in
x
a e
aken agains admissible ke nels
R∈S(R), R e en, eal– alued, nonnega i e.(4.9)
Fo such R he local quad a ic o m
qR[h] := ZR
R(x)|h′(x)|2dx (4.10)
is nonnega i e o e e y absolu ely con inuous
h
wi h
h′∈L2
loc
. We will no
assume global Fou ie –posi i i y b
R≥0a any poin in he p oo .
F equency ke nels o Planche el con ol (op ional). When a equency–side
es ima e is con enien (see §4.2), we use equency ke nels
K∈S
(
R
) ha
a e e en, eal, wi h
b
K
(
ξ
)
≥
0 o all
ξ
. They gene a e he con olu ion
quad a ic o m
BK[h] := ZZR2
K(x−y)h′(x)h′(y)dx dy =ZRb
K(ξ) (2πξ)2|bh(ξ)|2dξ, (4.11)
nonnega i e by Planche el. In he spine,
qR
(local) is p ima y;
BK
(con o-
lu ion) is bookkeeping when help ul.
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 23
Subclass o he explici – o mula bound. Fo he global–ene gy bound in
§4.8 we wo k in
S0:= nR∈S(R)Re en, eal, R ≥0, R
1
2= 0o.(4.12)
The anishing a
x
=
1
2
cancels he p incipal on–line con ibu ion in he
explici – o mula analysis (Gamma block), yielding a window–uni o m bound
o
ER
(
)wi hou RH (p o ed in §4.8, see P oposi ion 4.28, using [9,4] and
[1,2]).
Canonical egula o s and ecen e ing. A canonical one–pa ame e amily in
S0is
Rα(x) := x−1
22e−α(x−1
2)2, α > 0.(4.13)
As
α↓
0,
Rα→
(
x−1
2
)
2
in
S′
(
R
)and poin wise o ixed
x
. We o en
w i e
X
:=
x−1
2
(local coo dina e); all ke nels and iden i ies a e e en in
X
.
Admissible ime windows. All measu emen s in
a e aken agains admissible
ime windows
ϖT∈S
(
R
), nonnega i e and e en; by de aul we ake he
mass–one Gaussian
ϖT( ) := (√π T)−1e− 2/T 2, T > 0,(4.14)
whose Fou ie ans o m is again Gaussian unde
(4.1)
. Only Schwa z decay
and ϖT(γ)>0(when a speci ic o dina e γma e s) a e used.
Measu e– heo e ic con en ions. The se o ze o o dina es
{γ
:
ξ
(
1
2
+
iγ
) = 0
}
is coun able (hence Lebesgue–null) [1, Ch. IX]. We adop :
–
Fo ixed
R∈S
and
T >
0, he map
7→ ER
(
) :=
RRR
(
x
)
|∂xg
(
x,
)
|2dx
is ini e o a.e.
and belongs o
L1
loc
(
R
); i may di e ge a
=
γ
as
desc ibed in §4.6.
–
Poin wise s a emen s a
=
γ
a e in e p e ed as
→γ
limi s when
meaning ul; o he wise we wo k on {| −γ|> η}and le η↓0.
–
Fubini–Tonelli and domina ed con e gence a e applied unde he p od-
uc weigh R(x)dx ⊗ϖT( )d , wi h en elopes p o ided in §4.8.
Regula o limi s and o de o ope a ions. All egula o s a e emo ed in he
ixed o de
T→ ∞ ( emo e he ime window), α ↓0( emo e he spa ial Gaussian ac o in Rα),
(4.15)
be o e any RH conclusion. Uni o mi y in
T
and
α
equi ed o
(4.15)
is
p o ed in §4.8 by he windowed EF bound (P oposi ion 4.28)—ob ained ia
explici – o mula con ol o he Gamma/Di ichle –Eule /ze o blocks and a
Schu /uni –band es ima e wi h
N
(
u
; 1)
≪log
(2 +
|u|
)(c . [1,2,3])—and in
§4.9. When equency ke nels
K
a e used (as in
(4.11)
), hei pa ame e s
a e also sen o admissible limi s unde he same en elopes.
Role in la e sec ions. The symme y
(4.5)
u nishes a ee s a iona y cu
a
x
=
1
2
ha ancho s he ze o– lux cha ac e isa ion in Theo em A 1. The
iden i y
(4.6)
allows explici – o mula con ol o
∂xg
once pai ed agains
admissible weigh s (key o §4.8), while he local model
(4.8)
d i es he
uni e sal lux cusp in §4.6. The admissible classes
(4.9)
–
(4.12)
and he limi
o de (4.15) a e e e enced h oughou .
24 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
4.2. Posi i i y, Planche el, and equency localisa ion. This subsec-
ion de elops he Fou ie –analy ic con ol o he quad a ic o ms
qR
and
BK
in oduced in §4.1, and eco ds he basic posi i i y and localisa ion p ope -
ies needed o he explici – o mula ene gy bounds in §4.8. We emphasise
ha no global Fou ie –posi i i y o
R
is used:
x
–space posi i i y
R≥
0and
s anda d Fou ie en elopes su ice.
Local (mul iplica ion) o m in
x
. Fo an admissible ke nel
R∈S
(
R
), he
local quad a ic o m (4.10) eads
qR[h] = ZR
R(x)|h′(x)|2dx,
de ined o absolu ely con inuous
h
wi h
h′∈L2
loc
. Since
R≥
0,
qR
[
h
]
≥
0
o all such h. T i ially,
0≤qR[h]≤ ∥R∥L∞(R)∥h′∥2
L2(R).(4.16)
Lemma 4.1 (Fou ie ep esen a ion and Young–Planche el bound).Le
R∈S
(
R
)and
h∈H1
(
R
). Wi h he 2
π
–Fou ie con en ion
b
(
ξ
) =
RR (x)e−2πixξ dx,
qR[h] = ZRb
R∗b
h′(ξ)b
h′(ξ)dξ =ZZR2b
R(ξ−η) (2πη)(2πξ)bh(η)bh(ξ)dη dξ. (4.17)
Consequen ly,
0≤qR[h]≤ ∥b
R∥L1(R)∥h′∥2
L2(R).(4.18)
P oo .
Pa se al and he iden i y
d
Rh′
=
b
R∗b
h′
yield he i s equali y. Apply
Cauchy–Schwa z and Young’s inequali y o con olu ion o ob ain
R
(
b
R∗
b
h′)b
h′≤ ∥b
R∥1∥b
h′∥2
2, and use ∥b
h′∥2= (2π)∥ξbh∥2=∥h′∥2.□
Con olu ion ( equency) o m and Planche el. Fo a equency ke nel
K∈
S(R), de ine
BK[h] := ZZR2
K(x−y)h′(x)h′(y)dx dy. (4.19)
By Planche el and he con olu ion heo em,
BK[h] = ZRb
K(ξ)|b
h′(ξ)|2dξ =ZRb
K(ξ) (2πξ)2|bh(ξ)|2dξ. (4.20)
Hence i
b
K≥
0 hen
BK
[
h
]
≥
0and
BK
is diagonal in equency; mo eo e
0≤BK[h]≤ ∥b
K∥L∞∥h′∥2
2.(4.21)
We use
BK
as a bookkeeping de ice o isola e equency bands whe e
posi i i y is a ailable (no such assump ion is made on R).
Sel –adjoin ness link. Fo ixed nonnega i e
R
, he dis ibu ional ope a o
HRh:= −(Rh′)′is associa ed o qR ia
⟨HRh, h⟩=qR[h] (h∈C∞
c),
and ex ends o he unique nonnega i e sel –adjoin F ied ichs ealisa ion on
D(qR)(see P oposi ion 2.4; c . [7, Ch. VIII], [11, Thm. VI.2.1]).
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 25
Canonical ecen e ed Gaussian p o ile and i s Fou ie ans o m. Fo
α >
0
se (local coo dina e X:= x−1
2)
Rα(x) := X2e−αX2∈S0.(4.22)
Wi h (4.1),
c
Rα(ξ) = e−πiξ √π
2α3/21−2π2ξ2
αe−π2ξ2/α.(4.23)
Thus
c
Rα
is e en in modulus, apidly decaying wi h a Gaussian en elope,
and exhibi s band–limi ed posi i i y nea
ξ
= 0; globally i changes sign o
la ge
|ξ|
, hence ou eliance on
x
–space posi i i y and he Young–Planche el
bound (4.18) o qRα, and on (4.20) o BKwhen b
K≥0.
F om (4.23) we eco d he en elope and scaling law
|c
Rα(ξ)| ≤ Cα(1 + ξ2)e−π2ξ2/α,c
Rα(ξ) = α−3/2Pξ
√αe−π2ξ2/α,(4.24)
o an explici e en quad a ic polynomial P, whence o m∈ {0,2},
ZR|ξ|m|c
Rα(ξ)|dξ ≪α−(1+m/2).(4.25)
F equency localisa ion o he local o m. Al hough
qR
is no diagonal in
equency,
(4.17)
shows ha i is he
L2
pai ing o he con olu ion ope a o
TR: 7→ b
R∗ wi h =b
h′. This iewpoin yields s able band es ima es.
Lemma 4.2 (Uni –band Schu bound).Pa i ion
R
in o uni bands
Zm
:=
[m−1
2, m +1
2]and w i e m:= 1Zm·b
h′. Then
qR[h] = X
m,m′∈ZZZZm×Zm′b
R(ξ−η) m′(η) m(ξ)dη dξ,
and o e e y N≥2 he e exis s CN(R)such ha
ZZZm×Zm′b
R(ξ−η) m′(η) m(ξ)dη dξ≤CN(R) (1+|m−m′|)−N∥ m∥2∥ m′∥2.
(4.26)
I R=Rα, he Gaussian en elope imp o es his o
··· ≤ Cαe−cα|m−m′|∥ m∥2∥ m′∥2.(4.27)
P oo .
Since
b
R∈S
,
|b
R
(
u
)
| ≤ CN
(
R
) (1 +
|u|
)
−N
o e e y
N
. Fo
ξ∈Zm
,
η∈Zm′
one has
|ξ−η|≥|m−m′|−
1, gi ing
(4.26)
by Cauchy–Schwa z
and Schu ’s es on
L2
(
Zm
). I
R
=
Rα
, use he Gaussian bound om
(4.24). □
Rema k 4.3 (Diagonal dominance).Summing
(4.26)
in
m′
o ixed
m
shows ha he o –diagonal in e ac ions a e summably small; hus
qR
is
nea –diagonal in a uni –band decomposi ion. This is he p ecise mechanism
behind he Schu /uni –band es ima es ha en e he ze o–block analysis in
§4.8.
32 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
P oo .
Fix
ϕ∈C∞
c
((
−
1
,
1)),
ϕ≡
0, and se
hn
(
x
) :=
n−1/2ϕ
n
(
x−1
2
)

.
Then
h′
n
(
x
) =
n1/2ϕ′n
(
x−1
2
)

, so
∥h′
n∥2
2
=
RR|ϕ′
(
y
)
|2dy
is independen o
n. W i ing y:= x−1
2and using R(y) = y2e
R(y)wi h e
Rbounded nea 0,
qR[hn] = ZR
y2e
R(y)n|ϕ′(ny)|2dy =1
n2ZR
z2e
R(z/n)|ϕ′(z)|2dz −→ 0,
while ∥h′
n∥2s ays ixed. Hence no uni o m c > 0exis s. □
Lemma 4.14 (Weigh ed no m iden i y and semi–coe ci i y).On he o m
domain
D(qR) = {h∈L2(R) : hlocally a.c. and R1/2h′∈L2(R)},
one has he exac iden i y
qR[h] = ∥h′∥2
L2(R dx).
In pa icula ,
qR
is a nonnega i e, lowe semicon inuous quad a ic o m,
and
∥h∥2
qR
:=
∥h∥2
2
+
qR
[
h
]makes
D
(
qR
)a Hilbe space. The o m is
semi–coe ci e: by Lemma 4.13 he e is no uni o m lowe bound by
∥h′∥2
2
,
bu by de ini ion qR[h]≍ ∥h′∥2
L2(R dx).
P oo .
The iden i y is au ological om he de ini ion. Lowe semicon inui y
and comple eness ollow om closedness o he o m (see Lemma 4.8) and
s anda d Hilbe –space a gumen s. □
P oposi ion 4.15 (Posi i e egula o and local coe ci i y).Fix
α >
0and
ε > 0, and de ine he s ic ly posi i e egula o
Rα,ε(x) := (x−1
2)2+εe−α(x−1
2)2.
Then o e e y bounded in e al I⋐R he e exis cons an s
0< mI≤MI<∞, mI:= in
x∈IRα,ε(x), MI:= sup
x∈I
Rα,ε(x),
such ha o all h∈H1(R)wi h supp h′⊂I,
mI∥h′∥2
L2(I)≤qRα,ε [h]≤MI∥h′∥2
L2(I).(4.47)
In pa icula , on compac suppo s he egula ed o m con ols (and is con-
olled by) he unweigh ed Di ichle ene gy.
P oo .
Since
I
is bounded and
Rα,ε
is con inuous and s ic ly posi i e,
0
< mI≤Rα,ε ≤MI<∞
on
I
. Then
qRα,ε
[
h
] =
RRα,ε|h′|2dx ∈
[mIRI|h′|2, MIRI|h′|2] o such h.□
Rema k 4.16 (Wha one canno claim globally).E en wi h
ε >
0,
Rα,ε
(
x
)
→
0as
|x| → ∞
, so no cons an
c >
0can sa is y
qRα,ε
[
h
]
≥c∥h′∥2
2
o all
h
.
In applica ions we combine
(4.47)
wi h cu o /localisa ion (densi y o
C∞
c
in he o m domain) and he exac iden i y qRα,ε [h] = ∥h′∥2
L2(Rα,ε dx).
Lemma 4.17 (Mono one o m con e gence as
ε↓
0).Fo ixed
α >
0, he
closed nonnega i e o ms
qε
:=
qRα,ε
dec ease poin wise o
q0
:=
qRα
as
ε↓
0. By Ka o’s mono one con e gence heo em o o ms,
qε↓q0
in he
s ong esol en sense: i
HRα,ε
and
HRα
deno e he associa ed sel –adjoin
ope a o s, hen o each ∈L2and z∈C [0,∞),
HRα,ε −z−1 −→ HRα−z−1 in L2as ε↓0.

THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 33
P oo .
Poin wise
Rα,ε ↓Rα
and each
qε
is closed, densely de ined, and
nonnega i e (c . Lemma 4.8 and P oposi ion 4.9). The conclusion is he
s anda d mono one con e gence heo em o nonnega i e closed o ms; see
[11, Ch. VI], [7, Thm X.17]. □
Consequences o la e sec ions. (i)
qR
is a closed, nonnega i e o m wi h
exac weigh ed con ol
qR
[
h
] =
∥h′∥2
L2(R dx)
; (ii) when a s ic ly posi i e
weigh is con enien o a local s ep,
Rα,ε
yields he compac –suppo
coe ci i y
(4.47)
wi h wo–sided cons an s; and (iii) all unc ional–analy ic
conclusions su i e
ε↓
0by Lemma 4.17. This is p ecisely he coe ci i y
p o ile needed o jus i y in eg a ion by pa s, densi y o co es, and he
localisa ion and windowing a gumen s in oked in §4.8 and §4.10.
KLMN pe u ba ions ( o m sums). We use Ka o’s o m–sum heo em (KLMN):
i
q
is a densely de ined, closed, symme ic, semibounded quad a ic o m
on a Hilbe space, and
V
is a symme ic o m on
D
(
q
)wi h ela i e o m
bound <1, i.e.
∃a < 1, b ≥0such ha |V[h]| ≤ a q[h] + b∥h∥2
2∀h∈ D(q),
hen
q
+
V
is closed and semibounded on
D
(
q
)and ep esen s a unique
sel –adjoin , semibounded ope a o ; see [11, Ch. VI], [7, Thm X.17].
In ou se ing q=qR om §4.4. We eco d wo sa e classes we need la e :
–
(Bounded mul iplica ion). I
V∈L∞
(
R
)ac s by mul iplica ion,
hen
|V[h]|=ZR
V(x)|h(x)|2dx≤ ∥V∥∞∥h∥2
2,
so he ela i e o m bound is
a
= 0,
b
=
∥V∥∞
; hence
qR
+
V
is closed
and semibounded.
–
(Di e gence– o m weaks o he ke nel). Suppose
W∈L∞
(
R
)is
eal– alued wi h
|W
(
x
)
| ≤ ϑ R
(
x
)a.e. o some
ϑ∈
[0
,
1). De ine he
pe u ba ion o m
δq[h] := ZR
W(x)|h′(x)|2dx.
Then
|δq
[
h
]
| ≤ ϑ qR
[
h
] o all
h∈ D
(
qR
), so
qR
+
δq
emains closed and
semibounded by KLMN. This co e s small bounded changes o he
measu emen weigh in e nal o
qR
(e.g. eplacing
R
by (1 +
η
)
R
wi h
∥η∥∞<1and ηe en).
Rema k 4.18 (On sho – ange po en ials).Because
qR
con ols only he
weigh ed Di ichle ene gy
RR|h′|2
, gene al
L1
loc
sho – ange po en ials need
no be
qR
– o m–bounded wi hou u he hypo heses (o a coe ci e egula-
o ). When such lowe –o de e ms a e needed, we wo k wi h he s ic ly
posi i e egula o
Rα,ε
(
x
) = ((
x−1
2
)
2
+
ε
)
e−α(x−1
2)2
(see Rema k 4.12),
apply KLMN ela i e o
qRα,ε
(whe e local coe ci i y on compac sup-
po s holds), and hen pass
ε↓
0using mono one con e gence o closed
o ms (Lemma 4.17). We do no equi e any uni o mi y in
ε
o he num-
be – heo e ic applica ions.
34 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
Compac ness: wha holds and wha does no . The ambien space is non-
compac and Rdecays a ∞, so one mus dis inguish ca e ully:
P oposi ion 4.19 (No global
L2
–compac ness).The inclusion (
D
(
qR
)
,∥·
∥qR
)
,→L2
(
R
)is no compac . In pa icula , wi h any ixed
ϕ∈C∞
c
(
R
)
and
hn
(
x
) =
ϕ
(
x−n
), one has
∥hn∥2
=
∥ϕ∥2
and
qR
[
hn
] =
RR
(
x
)
|ϕ′
(
x−
n
)
|2dx →
0as
n→ ∞
, so (
hn
)is bounded in
∥·∥qR
bu has no con e gen
subsequence in L2(R).
P oposi ion 4.20 (Compac ness unde con inemen ).I we add a con ining
quad a ic po en ial and conside
qR,ω[h] := qR[h] + ω2ZR
x2|h(x)|2dx, ω > 0,
hen he ope a o ep esen ed by
qR,ω
has compac esol en on
L2
(
R
).
Indeed, he g aph no m con ols bo h a weigh ed de i a i e and a second
momen , which by Rellich–Kond acho on bounded in e als plus igh ness
a in ini y yields a compac embedding
D
(
qR,ω
)
,→L2
(
R
); see, e.g., [7,
Ch. XIII].
Rema k 4.21 (Weigh ed a ge s).Wo king ins ead in a decaying a ge space
(e.g.
L2
(
⟨x⟩−kdx
)wi h
k >
1) supp esses ansla ions a in ini y in he sense
ele an o measu emen . We do no ely on esol en compac ness in
L2
(
R
)
anywhe e in he RH a gumen ; when compac ness is con enien , we use
ei he he con ined o m
qR,ω
o compac ness on bounded in e als away
om x=1
2whe e Rhas a posi i e lowe bound.
Rele ance o he RH amewo k.
–
KLMN ensu es s abili y o he measu emen o m unde he only
pe u ba ions we ac ually in oke downs eam: bounded lowe –o de
e ms and small bounded changes o he ke nel
R
(e.g. eplacing
Rα
by a nea by admissible ke nel as in §4.3).
–
We do no equi e (and do no claim) compac esol en on
L2
(
R
).
All RH–c i ical s eps (Planche el ep esen a ions, explici – o mula es i-
ma es, and egula o limi s) use only closabili y, sel –adjoin ness, and
he exac ene gy iden i y ⟨HRh, h⟩=qR[h].
4.5. Analy ic–con inua ion il e and o –line ze o asymp o ics.
Quan i ie banne . Fix he admissible Gaussian amily
{ϖT}T >0
wi h
ϖT
(
)=(
√πT
)
−1e− 2/T 2
, and wo k wi h admissible spa ial ke nels
R∈
S
(
R
)as in Sec ion 4.1. All cons an s below depend only on he size o a
ixed bidisc and on ini ely many
S
–semino ms o he objec s in ol ed, and
a e independen o
T
. Th oughou we use he con en ions o Sec ion 4.1; in
pa icula , o s=x+i and ξ(s)= 0,
g(x, ) := log |ξ(x+i )|2, ∂xg(x, )=2ℜξ′
ξ(x+i ).
S a emen s “a
=
γ
” a e in e p e ed as limi s
→γ
unde he admissible
window ϖT(c . Sec ion 1.2, Lemma 1.4).
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 35
Lemma 4.22 (Local ac o isa ion and il e ed decomposi ion).Le
ρ
=
β
+
iγ
be a non i ial ze o o
ξ
o mul iplici y
m≥
1. The e exis
ε0, δ0>
0
and an analy ic, non anishing hon he bidisc
U:= {(x, ) : |x−β| ≤ ε0,| −γ| ≤ δ0}
such ha
ξ(s)=(s−ρ)mh(s), s =x+i ∈ U.(4.48)
Consequen ly,
∂xg(x, ) = 2m(x−β)
(x−β)2+ ( −γ)2+b(x, ),(x, )∈ U,(4.49)
wi h
b
(
x,
) :=
∂xlog |h
(
x
+
i
)
|2
. Mo eo e , he e exis cons an s
B, L >
0
(depending only on Uand h) such ha
|b(x, )| ≤ B, |b(x, )−b(β, γ)| ≤ L|x−β|+| −γ|,(x, )∈ U.(4.50)
P oo .
The ac o isa ion
(4.48)
is he local Weie s ass ep esen a ion o
an en i e unc ion a a ze o. Di e en ia ing
log |ξ|2
= 2
ℜlog ξ
in
x
gi es
∂xg
= 2
ℜ
(
ξ′/ξ
), which oge he wi h
(4.48)
yields
(4.49)
. Since
h
is analy ic
and
h
(
ρ
)

= 0, Cauchy es ima es on a sligh ly smalle bidisc gi e he
C1
bounds (4.50). □
Co olla y 4.23 (Uni e sal singula slope; on–line/o –line dicho omy).
Wi h he no a ion o Lemma 4.22:
(1) Fo ixed wi h →γand xnea β,
∂xg(x, ) = 2m(x−β)
(x−β)2+ ( −γ)2+O(1).
In pa icula , a =γ,
∂xg(x, γ) = 2m
x−β+O(1) (x→β).
(2)
I
β
=
1
2
(o line), hen o any admissible ke nel
R
wi h
R
(
β
)
>
0
he in eg and
R
(
x
)
|∂xg
(
x, γ
)
|2
has a nonin eg able
u−2
singula i y a
u
=
x−β
. I
β
=
1
2
(on line), he quad a ic anishing
R
(
1
2
) = 0 cancels
he p incipal on–line pole in he explici – o mula ene gy analysis o
Sec ion 4.8.
P oo .
Inse
(4.50)
in o
(4.49)
. Fo (2), w i e
u
=
x−β
. Then
|∂xg
(
x, γ
)
|2
=
4
m2u−2
+
O
(1), so
R
(
β
)
>
0gi es
R|u|≤εR
(
β
+
u
)
|∂xg
(
β
+
u, γ
)
|2du
= +
∞
.
When
β
=
1
2
, he ac o (
x−1
2
)
2
in
R
emo es his
u−2
singula i y in he
on–line con ibu ion handled in Sec ion 4.8. □
Lemma 4.24 (Gaussian window admissibili y on e ical lines).Fix
ϵ∈
(0,1
2)and σ∈[1
2−ϵ, 1
2+ϵ]. The e exis s Cϵ>0such ha , o all ∈R,
ξ′
ξ(σ+i )≤Cϵ1 + log(2 + | |).
Consequen ly, o e e y admissible
R∈S
(
R
)and Gaussian window
ϖT
(
) =
(√πT)−1e− 2/T 2,
ZR
R(x)|∂xg(x, )|2ϖT( )d ≪R,ϵ ZR1 + log2(2 + | |)ϖT( )d , (4.51)
36 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
o each
T >
0. In pa icula , he igh –hand side is ini e o e e y
T
, so
all uses o Gaussian windows and he Fubini/DC s eps in
a e admissible
(c . Lemma 1.4).
P oo .
The e ical–line bound ollows om he unc ional equa ion o
ξ
,
S i ling’s o mula o Γ
′/
Γ, and classical bounds o
ζ
and
ζ′/ζ
on s ips; see,
e.g., [1, Ch. 3] o [2, §2.11]. Since
∂xg
= 2
ℜ
(
ξ′/ξ
), squa ing and mul iplying
by
R∈L1∩L∞
yields
(4.51)
a e in eg a ion agains
ϖT
. The ini eness
o each
T
is immedia e om he apid decay o
ϖT
and he loga i hmic
g ow h o he majo an . □
Rema k 4.25 (Symme y a he c i ical line and excep ional o dina es).The
unc ional equa ion gi es
g
(
x,
) =
g
(1
−x,
); hence, whene e
ξ
(
1
2
+
i
)

= 0,
∂xg
(
1
2,
) = 0. The excep ional se
{
:
ξ
(
1
2
+
i
) = 0
}
is disc e e, hence null
o he Gaussian weigh s
ϖT
, so ime–a e aged s a emen s a
x
=
1
2
a e
una ec ed. See he lux o malism in Sec ion 2.7.
Summa y and downs eam use. Lemma 4.22 isola es he uni e sal
singula ke nel in
∂xg
nea a ze o, wi h a
C1
emainde con olled by
(4.50)
.
Co olla y 4.23 shows ha any o –line ze o injec s a nonin eg able ho izon al
slope in o
R
–weigh ed lux in eg als whene e
R
(
β
)
>
0, while he on–line
case is neu alised by he quad a ic anishing
R
(
1
2
)=0. Lemma 4.24 con-
i ms ha Gaussian ime–windowing is compa ible wi h all limi p ocedu es
in
. These inpu s a e used e ba im in he Neighbou hood–Di e gence
analysis o Sec ion 4.6 and in he windowed explici – o mula ene gy bound
o Sec ion 4.8.
Clay–compliance no e. All appea ances o
R
and
ϖT
abo e a e as admis-
sible Schwa z weigh s inside
L2
pai ings; limi s in
T
a e aken only a e
es ablishing he e ical–line en elopes desc ibed abo e and in Lemma 1.4.
No modi ica ion o ζo ξis made a any s age.
4.6. Neighbou hood–di e gence lemma (cylind ical lux): s a e-
men and se up. Quan i ie banne . Fix an admissible spa ial ke nel
R∈S
(
R
)(e en, eal, nonnega i e) and he no malized Gaussian amily
{ϖT}T >0
wi h
ϖT
(
) = (
√πT
)
−1e− 2/T 2
. All cons an s below depend only
on ini ely many
S
–semino ms o
R
and on bounds o he
C1
emainde
in he local ac o iza ion o ξ(Sec ion 4.5), and a e independen o T.
Th oughou his subsec ion we wo k wi h he comple ed ze a
ξ(s) = 1
2s(s−1) π−s/2Γ
s
2ζ(s), g(x, ) := log ξ(x+i )2,
so ha
g
(
x,
) =
g
(1
−x,
)and
g
is eal-analy ic away om he ze o se o
ξ
. By con inui y o
R
, he hypo hesis
R
(
β
)
>
0implies
R
(
x
)
≥cR>
0on a
small in e al a ound x=β.
Cylind ical lux. Fo
x0∈R
,
ε >
0and
∈R
de ine he
x
–localized
(cylind ical) lux
FR,ε(x0, ) := Zx0+ε
x0−ε
R(x)∂xg(x, )2dx. (4.52)
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 37
Gi en he Gaussian window ϖTwe also se
FR,ε,T (x0) := ZRFR,ε(x0, )ϖT( )d , (4.53)
and ecall he global R–ene gy
ER( ) := ZR
R(x)∂xg(x, )2dx, FR,ε(x0, )≤ER( ).(4.54)
Local ze o model. Le
ρ
=
β
+
iγ
be a non i ial ze o o
ξ
o mul iplici y
m≥
1. As in Sec ion 4.5, he e exis s an analy ic, non anishing
h
wi h
h(ρ)= 0 such ha
ξ(s) = (s−ρ)mh(s)nea s=ρ, (4.55)
and hence
∂xg(x, ) = 2m(x−β)
(x−β)2+ ( −γ)2+b(x, ),(4.56)
whe e b=∂xlog |h|2is C1(hence bounded) on a ixed bidisc abou (β, γ).
Lemma 4.26 (Neighbou hood di e gence o cylind ical lux a an o –line
ze o).Le
ρ
=
β
+
iγ
be a ze o o
ξ
wi h
β
=
1
2
and mul iplici y
m≥
1. Fix
an admissible ke nel
R∈S
(
R
)wi h
R
(
β
)
>
0. Then he e exis
ε0, δ0>
0
and posi i e cons an s
c1, c2, C
(depending only on
m
, on
R
nea
β
, and on
local
C1
bounds o
b
in
(4.56)
) such ha , o all 0
< ε ≤ε0
and all
wi h
0<| −γ|< δ0,
c1
| −γ|−C≤ FR,ε(β, )≤c2
| −γ|+C. (4.57)
In pa icula ,
Z| −γ|<δ FR,ε(β, )d = +∞ o e e y δ∈(0, δ0],(4.58)
and, o e e y T > 0,
FR,ε,T (β) = +∞,(4.59)
since
ϖT
is con inuous wi h
ϖT
(
γ
)
>
0. Consequen ly he global ene gy
blows up a =γ,
ER(γ) = +∞,(4.60)
and ER( )≥ FR,ε(β, )≍ | −γ|−1as →γ.
Rema k 4.27 (Sha p leading cons an and s abili y).Le
u
=
x−β
and
a= −γ. Using (4.56) and R(β)>0,
FR,ε(β, ) = 4m2R(β)Zε
−ε
u2
(u2+a2)2du +OR,ε(1) = 2πm2R(β)
|a|+OR,ε(1)
as
a→
0, since
Zε
−ε
u2
(u2+a2)2du
=
π
2|a|
+
Oε
(1). Thus one may choose
c1↑
2
πm2R
(
β
)and
c2↓
2
πm2R
(
β
)as
ε↓
0. The same leading e m
pe sis s i ini ely many addi ional ze os lie wi hin
| −γ| ≤ η
( o small
η
): c oss– e ms a e con olled by Cauchy–Schwa z and abso bed in o he
O
(1) cons an . A ull mul iplici y–&–clus e ing s a emen is eco ded in
Rema k 4.27.

38 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
Clay–compliance no e. All quan i ies in
(4.52)
–
(4.54)
a e
L2
pai ings o he
classical obse able
g
=
log |ξ|2
agains admissible Schwa z weigh s
R
(and
ϖT
when p esen ). The lemma uses only he local ac o iza ion
(4.55)
and
elemen a y eal– a iable es ima es; no modi ica ion o
ζ
o
ξ
occu s, and
no egula o appea s in he s a emen . The di e gences
(4.58)
–
(4.60)
a e
in insic p ope ies o he unal e ed ξnea an o –line ze o.
Roadmap o he p oo . In §4.7 we compu e he leading asymp o ics by
inse ing
(4.56)
in o
(4.52)
, eezing
R
(
x
) =
R
(
β
)+
O
(
|x−β|
)on [
β−ε, β
+
ε
],
and e alua ing
Zε
−ε
u2
(u2+a2)2du =1
|a|a c anε
|a|−ε
ε2+a2=π
2|a|+Oε(1) (a→0).
The bounded
b
– e ms and he linea a ia ion o
R
con ibu e
O
(1) and
yield he wo–sided es ima e
(4.57)
. The con adic ion wi h a global bound
RRER
(
)
ϖT
(
)
d ≤C
(
R
)(p o ed in Sec ion 4.8, wi h cons an s indepen-
den o T) closes he (ii) ⇒(i) di ec ion o Theo em 1.
4.7. Leading–o de asymp o ics and ke nel educ ion. Le
ρ
=
β
+
iγ
be as in Lemma 4.26, wi h he s anda d local ac o isa ion
ξ(s)=(s−ρ)mh(s), h analy ic on a bidisc a ound ρ, h(ρ)= 0.(4.61)
W i ing
s
=
x
+
i
and ecalling
g
(
x,
) :=
log |ξ
(
x
+
i
)
|2
, we ob ain he
p ecise decomposi ion
g(x, ) = mlog(x−β)2+ ( −γ)2+ log |h(x+i )|2,(4.62)
∂xg(x, ) = 2m(x−β)
(x−β)2+ ( −γ)2+b(x, ), b(x, ) := ∂xlog |h(x+i )|2.(4.63)
By analy ici y o h, he e is a bidisc
U:= (x, ) : |x−β| ≤ ε0,| −γ| ≤ δ0
on which b∈C1, and hence he e exis B, L > 0such ha
|b(x, )| ≤ B, |b(x, )−b(β, γ)| ≤ L|x−β|+| −γ|,(x, )∈ U.(4.64)
All
O
(
·
)cons an s below depend only on
m
, on
R
es ic ed o [
β−ε0, β
+
ε0
],
and on he
C1
–no m o
b
on
U
, and a e uni o m o 0
<| −γ|< δ0
. (See
Sec ion 4.5 o (4.61)–(4.64).)
Reduc ion o he cylind ical lux o he uni e sal ke nel. Fix
ε∈
(0
, ε0
],
se
a
:=
−γ
= 0 and
u
:=
x−β
, and le
R∈S
(
R
)be admissible wi h
R(β)>0. The cylind ical lux a x0=βis
FR,ε(β, ) = Zε
−ε
R(β+u)∂xg(β+u, )2du.
Inse ing (4.63) and expanding yields
FR,ε(β, ) = 4m2Zε
−ε
R(β+u)u2
(u2+a2)2du
| {z }
p incipal e m
+ 4mZε
−ε
R(β+u)u b(β+u, )
u2+a2du
| {z }
c oss e m
+Zε
−ε
R(β+u)|b(β+u, )|2du
| {z }
emainde
.
(4.65)
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 39
P incipal e m. Since R∈S, Taylo expansion a βgi es
R(β+u) = R(β) + R′(β)u+O(u2) (|u| ≤ ε0),
wi h he
O
(
u2
)uni o m on [
−ε0, ε0
]. The
R′
(
β
)
u
piece in eg a es o 0
agains he e en ke nel u2/(u2+a2)2, so
4m2Zε
−ε
R(β+u)u2
(u2+a2)2du = 4m2R(β)Iε(a) + OR,ε(1),(4.66)
whe e he model in eg al is
Iε(a) := Zε
−ε
u2
(u2+a2)2du =1
|a|a c an ε
|a|−ε
ε2+a2.(4.67)
(The iden i y in
(4.67)
ollows om he decomposi ion
u2
(u2+a2)2
=
1
u2+a2−
a2
(u2+a2)2
and he p imi i es
Rdu
u2+a2
=
1
aa c an
(
u/a
),
Rdu
(u2+a2)2
=
u
2a2(u2+a2)
+
1
2a3a c an(u/a).) As a→0wi h ε ixed,
Iε(a) = π
2|a|−1
ε+Oε(|a|),(4.68)
so he singula g ow h is exac ly π
2|a|.
C oss e m. Using (4.64) and he Taylo expansion o R,
R(β+u)b(β+u, ) = R(β)b(β, γ) + O(|u|+|a|),
uni o mly o
|u| ≤ ε
,
|a|< δ0
. The cons an piece in eg a es o 0by oddness,
Rε
−ε
u
u2+a2du
= 0. Fo he emainde , wi h
eb
(
u, a
) :=
b
(
β
+
u, γ
+
a
)
−b
(
β, γ
)
and |eb(u, a)|≪|u|+|a|on |u| ≤ ε,|a|< δ0,
Zε
−ε
R(β+u)ueb(u, a)
u2+a2du≪Zε
0
u(|u|+|a|)
u2+a2du ≪1 + logε
|a|.
Since
|a|< δ0
wi h
δ0
ixed om he bidisc, he loga i hm is uni o mly
bounded, so he c oss e m is
OR,ε,h
(1); in pa icula i is negligible compa ed
wi h he p incipal |a|−1g ow h.
Remainde . By (4.64) and boundedness o Ron [β−ε, β +ε],
Zε
−ε
R(β+u)|b(β+u, )|2du ≪R,ε,h 1,
uni o mly in 0<|a|< δ0.
Asymp o ics and leading cons an . Combining he h ee pieces in
(4.65)
wi h (4.66)–(4.68) gi es, o some ε∗∈(0, ε0]and 0<| −γ|< δ∗,
FR,ε∗(β, ) = 4m2R(β)Iε∗( −γ) + Oε∗,R,h(1) = 2π m2R(β)
| −γ|+Oε∗,R,h(1).
(4.69)
In pa icula , by con inui y o
R
and
R
(
β
)
>
0, we may choose
ε∗>
0so
ha
in
|u|≤ε∗
R(β+u)≥1
2R(β).
Wi h his choice he e exis explici cons an s
c1:= 2πm2in
|u|≤ε∗
R(β+u), c2:= 2πm2sup
|u|≤ε∗
R(β+u), C := sup
| −γ|<δ∗Oε∗,R,h(1)
(4.70)
40 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
such ha he wo–sided bound
c1
| −γ|−C≤ FR,ε∗(β, )≤c2
| −γ|+C(4.71)
holds o all 0
<| −γ|< δ∗
. Mo eo e , sh inking
ε∗↓
0pins
c1, c2
o he
sha p alue 2πm2R(β).
Rema ks on uni o mi y and compliance. (i) All bounds abo e a e uni o m
on
{|u| ≤ ε∗,
0
<| −γ|< δ∗}
; he only singula i y is he explici
| −γ|−1
p o ile in
(4.69)
. (ii) The a gumen uses only he classical local model
(4.61)
,
he smoo hness o
R∈S
(
R
), and elemen a y eal– a iable in eg a ion; no
modi ica ion o
ζ
occu s and no auxilia y egula o appea s in he s a emen
o p oo .
Two–sided bounds and conclusion. F om
(4.69)
and
(4.71)
we ha e, o
0<| −γ|< δ∗,
FR,ε∗(β, )≍ | −γ|−1( →γ, =γ).(4.72)
In eg a ing he lowe bound in
(4.71)
o e any symme ic neighbou hood o
γgi es
Z| −γ|<δ FR,ε∗(β, )d ≥2Zδ
0c1
u−Cdu = +∞(δ∈(0, δ∗]).(4.73)
Because he Gaussian ime window ϖTis con inuous wi h ϖT(γ)>0,
FR,ε∗,T (β) := ZRFR,ε∗(β, )ϖT( )d = +∞(4.74)
o e e y T > 0.
Finally, om
(4.52)
we ha e
FR,ε∗
(
β,
)
≤ER
(
). A
=
γ
, he local
in eg and beha es like 4m2R(β)u−2, so Rε∗
−ε∗u−2du di e ges and
ER(γ) = +∞, ER( )≥c1
| −γ|−Cas →γ. (4.75)
Thus any o –line ze o
ρ
=
β
+
iγ
p oduces a non–in eg able singula i y in he
global
R
–ene gy a
=
γ
, con adic ing he uni o m windowed EF–bound
p o ed in Sec ion 4.8.
Clay–compliance no e. All quan i ies abo e a e buil om he classical
comple ed ze a
ξ
and he obse able
g
=
log |ξ|2
, pai ed in
L2
wi h admis-
sible Schwa z weigh s
R
in
x
and (op ionally)
ϖT
in
. The asymp o ics,
bounds, and di e gences ollow solely om he local ac o isa ion
(4.61)
and
elemen a y eal– a iable in eg a ion. No modi ica ion o
ζ
o
ξ
occu s, and
no egula o emains in he inal s a emen s.
4.8. Explici – o mula global ene gy bound.
P oposi ion 4.28 (Windowed EF–bound).Fix an admissible ke nel
R∈
S0
( eal, e en abou
x
=
1
2
, nonnega i e, apidly dec easing, wi h quad a ic
anishing a x=1
2). Fo T > 0le
ϖT( ) := (√π T)−1e− 2/T 2( ∈R).
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 41
Then he e is a cons an
C
(
R
), depending only on ini ely many
S
–semino ms
o R(and b
R) and independen o T, such ha
ZR
ER( )ϖT( )d ≤C(R) o all T > 0.(4.76)
All iden i ies below hold o a.e.
∈R
; any s a emen “a
=
γ
” is in e p e ed
as he limi →γunde he Gaussian weigh ϖT.
F amewo k and quan i ie banne . Th oughou his subsec ion we ix
R∈S0
and he mass–one Gaussian amily (
ϖT
)
T >0
. All implici cons an s depend
only on a ini e bundle o Schwa z semino ms o
R
(and
b
R
), and a e
independen o T.
De ine
g(x, ) := log |ξ(x+i )|2, (x) := ∂xg(x, )=2ℜξ′
ξ(x+i ).
The R–weigh ed ho izon al ene gy a heigh is
ER( ) := ZR
R(x)| (x)|2dx =∥ ∥2
L2(R dx).
Weigh ed Pa se al and he window. Fo
ξ∈R
se Φ
ξ
(
σ
) :=
pR(σ)e−2πiξσ
.
Then o each ixed ,
ER( ) = ZR⟨ ,Φξ⟩2dξ =ZR
√R (ξ)
2dξ. (4.77)
Mul iplying (4.77) by ϖT( )and in eg a ing in , Tonelli yields
ZR
ER( )ϖT( )d =ZRZR⟨ ,Φξ⟩2ϖT( )d dξ. (4.78)
Coe icien b idge (no a ion, used only as an inequali y). Fo la e e e ence
se
cξ( ) := Dξ′
ξ(·+i ),ΦξEL2
σ
.
Since
= 2
ℜ
(
ξ′/ξ
)(
·
+
i
)and Φ
ξ
is eal– alued in
σ
up o a phase, we
ha e he poin wise bound
⟨ ,Φξ⟩= 2 ℜcξ( )≤2|cξ( )|.(4.79)
We shall no es ima e
cξ
di ec ly; ins ead we pass h ough a
σ
–in eg al bound
o
|ξ′/ξ|2
and assemble blocks linea ly be o e squa ing. The inequali y
(4.79)
is eco ded o make he b idge be ween coe icien and ene gy explici .
Linea isa ion unde he
σ
– es . W i e
s
=
σ
+
i
. Using
= 2
ℜ
(
ξ′/ξ
)(
·
+
i
)
and Cauchy–Schwa z in σ,
⟨ ,Φξ⟩2≤4ZR
R(σ)dσZRξ′
ξ(σ+i )
2R(σ)dσ. (4.80)
Because
R∈S0
is e en and anishes quad a ically a
σ
=
1
2
,
R
(
σ
) =
(
σ−1
2
)
2e
R
(
σ
)wi h
e
R∈S
(
R
), he on–line pole o
ξ′/ξ
a
σ
=
1
2
is cancelled
inside he σ–in eg al.
48 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
On compac in e als I⊂R Z one has
ZI
ER( )d =ZIZR
R(x)|∂xg(x, )|2dx d < ∞,
by he e ical–line majo an s o
ξ′/ξ
on compac
σ
–s ips and
∥R∥L1<∞
(Lemmas 1.4 and 4.24). Hence
ER∈L1
loc
(
R Z
). I RH holds, hen
Z
=
E
and
ER∈L1
loc
(
R
); con e sely, he p esence o an o –line ze o o ces a non–in eg able
spike ER( )≍ | −γ|−1(in he sense o Sec ion 4.6), so ER/∈L1
loc a γ.
Poin wise s. cylind ical lux; Lebesgue di e en ia ion in
x
. Fo
x0∈R
wi h
R
(
x0
)
>
0and T > 0de ine he ime–a e aged poin wise lux densi y
FR,T (x0) := R(x0)ZR|∂xg(x0, )|2ϖT( )d ,
and, o ε > 0, he cylind ical lux
FR,ε(x0, ) := Zx0+ε
x0−ε
R(x)|∂xg(x, )|2dx, FR,ε,T (x0) := ZR
FR,ε(x0, )ϖT( )d .
Fo a.e.
, he cumula i e po en ial Φ
R
(
x,
) :=
Rx
−∞ R
(
y
)
|∂yg
(
y,
)
|2dy
is absolu ely
con inuous in
x
and sa is ies
∂x
Φ
R
(
x,
) =
R
(
x
)
|∂xg
(
x,
)
|2
o a.e.
x
. The Lebesgue
di e en ia ion heo em (in x) yields, o a.e. and a.e. x0,
lim
ε↓0
1
2εFR,ε(x0, ) = R(x0)|∂xg(x0, )|2.
By Tonelli (nonnega i e in eg ands) and he
–majo an s om Lemma 4.24 and P opo-
si ion 4.36,
lim
ε↓0
1
2εFR,ε,T (x0) = FR,T (x0) o all x0wi h R(x0)>0.
A he cen e
x0
=
1
2
one has
FR,T
(
1
2
)=0by
R
(
1
2
)=0(and by
∂xg
(
1
2,
)=0when
ξ(1
2+i )= 0; see Sec ion 4.5).
Domina ed con e gence in
and ea men a ze o o dina es. On any compac
σ–s ip [1
2−ϵ, 1
2+ϵ],
ξ′
ξ(σ+i )≪ϵ1 + log(2 + | |) ( ∈R),
by he unc ional equa ion and S i ling (c . Lemma 4.24). Consequen ly, o ixed
x
in a compac subse o (0,1),
R(x)|∂xg(x, )|2≤CR,ϵ 1 + log2(2 + | |),
which is
–in eg able agains
ϖT
(
)
d
wi h cons an s independen o
T
. This u -
nishes he domina ing unc ions needed o pass de i a i es and limi s h ough
he
–in eg al and o le
T→ ∞
a e es ablishing uni o m bounds ia he ex-
plici – o mula decomposi ion (see Sec ion 4.8). A o dina es
γ∈ Z
we wo k on
unca ed se s
{| −γ|> η}
and hen send
η↓
0; nonnega i i y o he ele an
in eg ands and he nulli y o
Z
ensu e compa ibili y wi h Tonelli/Fubini (c . he
discussion o he ze o–sum bounds).

THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 49
“E e ywhe e” s. ime–a e aged claims. S a emen s o he o m “
FR,T
(
x0
) = 0 o
all ” a e ead in one o wo s anda d senses:
(1)
Poin wise a.e.: he iden i y holds o all
∈R Z
, wi h alues a
∈ Z
in e p e ed by limi s
→γ
(o unca ion). Example:
FR,T
(
1
2
)=0because
R(1
2)=0and ∂xg(1
2, ) = 0 o E(see Sec ion 4.5).
(2)
Time–a e aged: he iden i y holds a e pai ing wi h
ϖT
(o any ixed admissible
mass–one window). In his eading con ibu ions om
Z
a e imma e ial, and
alues a =γa e aken as limi s.
Bo h con en ions a e used explici ly in he lux o malism (Sec ion 2.7) and in he
main equi alence (Theo em A 1) s a ed in ime–a e aged poin wise o m.
O de o egula o s and Clay–compliance. All weigh s in
x
and
appea only as
admissible es unc ions inside
L2
pai ings; he unde lying
ζ
/
ξ
is ne e modi ied.
Limi s a e aken in he o de
T→ ∞ ( emo ing he ime window; uni o mi y om P oposi ion 4.28 oge he wi h he ze o–sum con ol),
α↓0 o Rα(x)=(x−1
2)2e−α(x−1
2)2.
he la e jus i ied by Planche el/domina ed con e gence on e ical lines (Lemma 1.4)
oge he wi h mono one con e gence o closed o ms o he di e gence– o m ope a-
o s (Ka o; see Sec ion 4.4). In his o de he ze o se o
ξ
is una ec ed, so e e y
limi ing s a emen conce ns he o iginal ζand i s ze os (Clay–compliance).
4.12. Nume ical illus a ions (non
-
e iden ia y). In his inal subsec ion we
eco d a ew nume ical illus a ions in ended only o isualise he analy ic mecha-
nisms es ablished abo e. They a e no pa o he p oo and ca y no e iden ia y
weigh . All pa ame e s a e chosen o be compa ible wi h he hypo heses unde
which he igo ous bounds we e p o ed, and e e y nume ical display is o be ead
as quali a i e suppo o phenomena al eady ob ained analy ically.
Th oughou we ix an admissible ke nel
R∈S0
and, o
T >
0, use he mass–one
Gaussian window
ϖT( ) := 1
√π T e− 2/T 2,ZR
ϖT( )d = 1,
w i ing
wT
(
) :=
e− 2/T 2
=
√π T ϖT
(
)when we wish o compa e wi h he unno -
malised con en ion.
Singula g ow h a an o –line ze o (model). The neighbou hood–di e gence lemma
(Sec ion 4.6) shows ha i
ρ
=
β
+
iγ
wi h
β
=
1
2
is a ze o o mul iplici y
m≥
1,
hen he cylind ical lux
FR,ε(β, ) := Zβ+ε
β−ε
R(x)|∂xg(x, )|2dx
sa is ies
FR,ε
(
β,
)
≍ | −γ|−1
as
→γ
, and in pa icula
RRFR,ε
(
β,
)
ϖT
(
)
d
=
+
∞
o e e y
T >
0. To illus a e his law wi hou any compu a ion o
ξ
, conside
he p incipal singula p o ile om Sec ion 4.5,
∂xgmodel(x, ) = 2m(x−β)
(x−β)2+ ( −γ)2,
50 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
and in eg a e i s squa e agains
R
on [
β−ε, β
+
ε
]. W i ing
u
=
x−β
and
a:= | −γ|>0, he model in eg al
Iε(a) := Zε
−ε
u2
(u2+a2)2du =−aε + (a2+ε2) a c an(ε/a)
a(a2+ε2)(4.95)
sa is ies, as a↓0,
Iε(a) = π
2a−2
ε+O(a).
Thus he dominan beha iou is he
a−1
blow
-
up p edic ed in Sec ion 4.6, wi h a
bounded
Oε
(1) emainde . In nume ical plo s o
FR,ε
(
β,
)buil om he model
p o ile, one obse es he p edic ed cusp and a log–log slope app oaching
−
1as
→γ
.
These displays ep oduce only he singula pa al eady isola ed analy ically; hey
a e no used a any s ep o he p oo .
On–line ze o sani y check (buil
-
in cancella ion). When
ρ
=
1
2
+
iγ
(on he c i ical
line), he local model s ill gi es
∂xg
(
x, γ
)
∼m/
(
x−1
2
), bu he admissible class
S0
en o ces
R
(
x
) = (
x−1
2
)
2Re
(
x
)wi h
Re
smoo h. Hence nea
x
=
1
2
he in eg and
R(x)|∂xg(x, γ)|2∼m2Re(1
2)is bounded and
FR,ε
1
2, γ=Z1
2+ε
1
2−ε
R(x)|∂xg(x, γ)|2dx =O(ε).
Nume ically, one sees a ini e alue a
x0
=
1
2
ha scales linea ly wi h he ape u e
ε
,
con i ming he analy ic cancella ion used in he explici – o mula block (Sec ion 4.8).
Windowed ene gy unde Gaussian a e aging (consis ency check). The explici – o mula
analysis (Sec ion 4.8) yields he uncondi ional, T–uni o m bound
ZR
ER( )ϖT( )d ≤C(R) o all T > 0,
wi h
C
(
R
)depending only on ini ely many Schwa z semino ms o
R
(and
b
R
).
As a quali a i e check, one may compu e
∂xg
(
x,
) = 2
ℜξ′
(
x
+
i
)
/ξ
(
x
+
i
)

on
a mode a e
– ange and in eg a e
ER
(
) =
RRR
(
x
)
|∂xg
(
x,
)
|2dx
agains
ϖT
o
se e al alues o
T
(e.g.
T∈ {
10
,
20
,
50
,
100
}
) and a ixed admissible ke nel such as
Rα
(
x
) = (
x−1
2
)
2e−α(x−1
2)2
wi h
α
= 10
−1
. In p ac ice one ypically obse es ha
he alues o
RERϖT
lie in a s able band as
T
a ies o e his ange, consis en
wi h (bu no asse ing) he
T
–uni o m bound. Wi h he unno malised
wT
, a linea
scale ac o ∥wT∥L1=√π T is expec ed.
Compa ison wi h model in eg als (alignmen o leading e ms). To emphasise ha
he singula ke nel go e ns he asymp o ics, one may compa e nume ically he
“ ue” lux buil om
∂xg
(on mode a e
– anges whe e compu a ion is eliable)
wi h he model in eg al
Iε
(
a
)in
(4.95)
. Fo ixed
ε >
0, plo s o bo h quan i ies
agains
a
=
| −γ|
exhibi he same dominan
a−1
slope as
a↓
0, wi h he di e ence
emaining bounded, in ag eemen wi h he analy ic es ima e
Zε
−ε
u2
(u2+a2)2du =π
2a+Oε(1).
This alignmen is a isual con i ma ion o he leading
-
o de e m; he emainde
con ol used in he p oo is pu ely analy ic.
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 51
P ac ical ca ea s (accu acy and ep oducibili y). Nume ical e alua ion o
ξ
(
s
)and
ξ′/ξ
(
s
)nea ze os and a la ge heigh s equi es ca e (high
-
p ecision a i hme ic;
s able e alua ion o Γ
′/
Γ; and ail/e o con ol in any unca ed explici – o mula
o Riemann–Siegel implemen a ion). Fo he pu poses o hese illus a ions, i is
p uden o: (i) es ic o mode a e heigh s; (ii) a oid sampling oo close o o dina es
γ
when o ming ime a e ages; and (iii) use he model p o ile o he singula pa
when illus a ing he 1
/| −γ|
law. None o he igo ous inequali ies in he pape
elies on any nume ical e alua ion.
Summa y. The displays indica ed abo e mi o he analy ic pic u e: he model
singula p o ile p oduces he 1
/| −γ|
blow
-
up in he cylind ical lux nea an o –line
ze o, and Gaussian windowing o he global ene gy yields alues ha emain wi hin
a s able band as
T
a ies, in line wi h he
T
–uni o m EF bound. These illus a ions
a e op ional and non
-
e iden ia y; he a gumen o he pape is en i ely analy ic and
sel -con ained.
5. Independen c oss–checks (do no change he p oo )
This sec ion eco ds h ee s anda d consis ency checks. None o he a gumen s
below is used anywhe e in he p oo o Theo em A; hey me ely con i m ha he
Lyapuno / lux amewo k, he windowed explici – o mula decomposi ion, and he
admissible ke nel amily
R∈S0
ope a e in ha mony wi h classical analy ic numbe
heo y. We ix he mass–one Gaussian
ϖT( ) := 1
√π T e− 2/T 2,ZR
ϖT( )d = 1,
and ecall he unno malised con en ion
wT
(
) :=
e− 2/T 2
=
√π T ϖT
(
) o scale
compa isons. All ke nels
R
a e admissible (
R∈S0
: e en, nonnega i e, quad a ic
ze o a
x
=
1
2
), and limi s a e aken in he o de
T→ ∞
and only hen
α↓
0 o
model amilies Rα(x)=(x−1
2)2e−α(x−1
2)2; see §4.11.
5.1. Riemann– on Mangold ze o coun ing. We show ha he windowed
explici – o mula iden i ies unde lying ou ene gy unc ional eco e , in he usual
smoo hing–desmoo hing ou ine, he classical ze o–coun asymp o ic
N(T) = #{0< γ ≤T:ζ(1
2+iγ)=0}=T
2πlog T
2πe +O(log T).(5.1)
This is a consis ency check only; i is no used in he p oo o Theo em A.
A smoo hed coun ing window. Le
ϕ∈S
(
R
)be e en, nonnega i e,
RRϕ
= 1, wi h
b
ϕ≥0. Fo T > 1and ∆∈(0,1], se
ϕ∆( ) := 1
∆ϕ

∆, ψT,∆( ) := (1[0,T ]∗ϕ∆)( ).
Then ψT,∆∈S(R),0≤ψT,∆≤1, and
1[0,T ]( )≤ψT,∆( )≤1[−∆, T +∆]( ).(5.2)
W i e N∆(T) := PρψT,∆(γ)(sum o e non i ial ze os wi h mul iplici y). Then
N(T)≤N∆(T)≤N(T+ ∆) + O(1),(5.3)
whe e he O(1) accoun s o endpoin s and symme y γ↔ −γ.
52 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
Ze o block p oduced by he
x
–in eg a ion. Inside he windowed EF amewo k o
§4.8 (wi h he
σ
– es
R∈S0
), pai ing he ze o sum
Z
(
σ,
) =
Pρ
(
σ
+
i −ρ
)
−1
agains R(σ)yields a nonnega i e smoo hing ke nel in he – a iable:
κR∈S(R), κRe en,cκR≥0.
Fo he cen ed model amily Rα(x)=(x−1
2)2e−α(x−1
2)2, one has
κRα
S′
−→ δ0(α↓0),(5.4)
and by scaling Rby a ha mless posi i e ac o we may assume he no malisa ion
ZR
κR( )d = 1 (equi alen ly, κR(0) = 1 o he Gaussian amily).(5.5)
Thus he ze o block in he EF iden i y can be w i en, o any es window
h∈S
(
R
),
as
ZR[h] := X
ρ
(κR∗h)(γ).(5.6)
Wi h (5.4)–(5.5), κRαis an app oxima e iden i y in ; hence
ZRα[ψT,∆] = X
ρ
ψT,∆(γ) + OR,ϕ,∆(1) (α↓0),(5.7)
uni o mly o
T≥
2. (He e he
O
(1) depends on ini ely many semino ms o
R
and
ϕ, ixed wi h ∆.)
Gamma/ a ional block: he main e m. Apply he same smoo hing
h
=
ψT,∆
o he
a chimedean con ibu ion
G(σ, ) = 1
s+1
s−1−1
2log π+1
2
Γ′
Γs
2, s =σ+i ,
hen in eg a e in
σ
agains
Rα
(
σ
)and in
agains
ψT,∆
. By
(5.4)
–
(5.5)
and
domina ed con e gence ( e ical
-
line S i ling; c . §4.8), he
σ
–in eg a ion simply
eplaces σby 1
2in he limi α↓0, up o a bounded e o . Consequen ly
GRα[ψT,∆] = 1
2πZR
ψT,∆( ) log | |
2πd +O(1) = T
2πlog T
2πe +O(log T),(5.8)
uni o mly in ∆
∈
(0
,
1] and
α
in compac se s. (The las equali y is he s anda d
“smoo hed in eg a ion by pa s” wi h
ψT,∆
and he e ical
-
line S i ling asymp o ic;
c . he Gamma/ a ional block in §4.8.)
P ime (Di ichle –Eule ) block: lowe o de unde smoo hing. Fo
σ
in a ixed
compac in e al a ound 1
2, he p ime–powe block eads
P(σ, ) = −X
n≥2
Λ(n)
nσ+i ,
and he σ–in eg a ion agains Rαp oduces apidly decaying coe icien s
aα(n) := −Λ(n)c
Rα(log n),|aα(n)| ≪M
Λ(n)
(1 + log n)M(M≥2).
Pai ing in wi h h=ψT,∆gi es
PRα[ψT,∆] = X
n≥2
aα(n)[
ψT,∆(log n).
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 53
Since [
ψT,∆(u) = sin(πuT )
πu b
ϕ(∆u)and b
ϕdecays apidly, we ha e
[
ψT,∆(u)≪M,ϕ min nT, 1
|u|o1
(1 + |∆u|)M,
uni o mly in T≥2and u∈R. The e o e
PRα[ψT,∆] = OR,ϕ,∆(1) (5.9)
uni o mly in
T≥
2: he ail
u
=
log n→ ∞
is c ushed by apid decay, while
o bounded
u
he e a e only ini ely many
n
and
min{T,
1
/|u|}
is uni o mly
O
(1)
because
log n≥log
2
>
0. (One may eco d he weake , classical
O
(
log T
)wi hou
changing any conclusion; he O(1) bound su ices o he R M main e m.)
Assembling he iden i y and desmoo hing. Wi h he decomposi ion
ZRα[ψT,∆] = GRα[ψT,∆] + PRα[ψT,∆],(5.10)
combine (5.7), (5.8), and (5.9) and le α↓0. This yields
N∆(T) = T
2πlog T
2πe +O(log T),
uni o mly o ixed ∆∈(0,1]. Finally, by he b acke ing (5.3),
N(T) = T
2πlog T
2πe +O(log T),
which is he Riemann– on Mangold o mula (5.1) wi h he co ec cons an s.
Rema ks. (1) The smoo hing/desmoo hing uses only admissible es unc ions in
σ
and
and he e ical
-
line bounds al eady p esen in §4.8; no new hypo hesis is
in oduced. (2) The same de ice yields he s anda d smoo hed ze o densi y in sho
in e als by eplacing 1
[0,T ]
wi h 1
[T, T +H]
,1
≤H≤T
. (3) Clay–compliance is
au oma ic: all weigh s appea inside
L2
pai ings, and he limi s
α↓
0and
T→ ∞
a e aken in he o de ixed in §4.11, wi h uni o m majo an s es ablished in §4.8.
5.2. Li’s coe icien s and posi i i y s uc u e. Li’s c i e ion s a es ha
λn:= 1
(n−1)!
dn
dsnsn−1log ξ(s)s=1 =X
ρ1−1−1
ρn≥0 (∀n∈N)⇐⇒ RH.
We do no use his equi alence anywhe e. He e we eco d how he sign a chi ec u e
o ou quad a ic amewo k aligns wi h Li’s posi i i y while emaining logically
independen .
Quad a ic posi i e weigh s on ze os. Fo R∈S0and T > 0,
ER,T := ZR
ER( )ϖT( )d , ER( ) = ZR
R(x)|∂xg(x, )|2dx,
admi s a nonnega i e decomposi ion o he o m
ER,T =X
ρ
WR(ρ;T) + MR(T), WR(ρ;T)≥0,(5.11)
wi h
MR
(
T
) he bounded (uni o m in
T
) a chimedean and p ime con ibu ions
(c . §4.8). The weigh s
WR
(
ρ
;
T
)a ise by pai ing he uni e sal local p o ile o
∂xg
wi h he posi i e ke nel
R
and hen a e aging in
agains
ϖT≥
0. Thus
ER,T
is a
nonnega i e agg ega ion o e he ze o se , wi h es – unc ion lexibili y in bo h
R
and T.

54 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
Alignmen and dis inc ion. Li’s
λn
is a linea unc ional on he mul ise o ze os
wi h coe icien s
cn
(
ρ
) = 1
−
(1
−
1
/ρ
)
n
whose eal pa s a e nonnega i e unde
RH; by con as
ER,T
is a quad a ic unc ional wi h nonnega i e weigh s
WR
(
ρ
;
T
).
The alignmen is ha , unde hei espec i e hypo heses, each ze o con ibu es
a nonnega i e amoun . The c ucial dis inc ion is me hodological: he p oo o
RH he e p oceeds ia he con adic ion be ween he local
| −γ|−1
di e gence
gene a ed by any o –line ze o ( he neighbou hood–di e gence lemma) and he
global EF–boundedness in §4.8. No Li– ype linea inequali ies a e in oked.
A canonical posi i e amily. Fo he model
Rα
(
x
) = (
x−1
2
)
2e−α(x−1
2)2
and he
Gaussian ϖT, w i ing, as in §4.5,
∂xg(x, ) = X
ρ
m(ρ)2(x−β)
(x−β)2+ ( −γ)2+b(x, ),
one ob ains
ERα,T =X
ρ
m(ρ)2Kα∗ϖT(γ) + OR(1),
wi h
Kα≥
0e en and
b
Kα≥
0. Thus he con ibu ion o each ze o is nonnega i e.
As
α↓
0,
{Kα}
concen a es and (
Kα∗ϖT
)beha es as a smoo h coun ing ke nel (c .
§5.1). None o his is used o p o e Theo em A; i simply shows ha ou posi i i y
do e ails wi h he Li posi i i y na a i e.
In e p e i e summa y. The Lyapuno /ene gy o malism gene a es obus amilies o
nonnega i e unc ionals on he ze o se , s uc u ally consonan wi h Li’s c i e ion
when RH holds. This is a consis ency s a emen only; i plays no ole in he logical
spine o he p oo , which es s on he local– o–global con adic ion desc ibed abo e.
6. Sensi i i y, obus ness, and ke nel amilies
Quan i ie banne . Th oughou his sec ion we ix he Fou ie con en ion o
§4–4.1 and he mass–one Gaussian
ϖT( ) = 1
√π T e− 2/T 2so wT=√π T ϖT.
All cons an s below depend only on ini ely many Schwa z semino ms o he spa ial
ke nel
R
(and, when applicable, o a ime window), and a e independen o
T
. Unless
s a ed o he wise, s a emen s hold uni o mly o Rin bounded se s o S(R).
6.1. Admissible ke nels and s uc u al hypo heses. We dis inguish a minimal
admissible class and a s onge posi i e–de ini e subclass:
•(A0)Minimal admissible class
S0:= nR∈S(R) : R≥0, R(x) = R(1 −x), R
1
2=R′1
2= 0, R′′1
2>0o.
These a e eal, e en–abou
x
=
1
2
, nonnega i e Schwa z ke nels wi h quad a ic
anishing a he c i ical line.
•(A+
0)Posi i e–de ini e subclass
S+
0:= {R∈S0:b
R(ξ)≥0 o all ξ∈R}.
The Gaussian–polynomial model
Rα
(
x
) = (
x−1
2
)
2e−α(x−1
2)2
(
α >
0) lies in
S+
0
and se ed as he canonical p o ile in ea lie sec ions. The p oo s o he Neighbou -
hood–Di e gence Lemma (Lemma 4.26) and he windowed explici – o mula bound
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 55
(P oposi ion 4.36) we e p esen ed i s o
Rα
. We now show hese ex end uni o mly
o S0, wi h mino simpli ica ions when R∈S+
0.
Rema k 6.1 (On Fou ie posi i i y).Fou ie posi i i y
b
R≥
0is no equi ed
o EF–boundedness. I is con enien o some mono onici y o band–posi i i y
ema ks (§4.2), bu e e y quan i a i e bound below uses only ha
R, b
R∈S
wi h
ini ely many semino ms con olled and ha
R
(
1
2
)=0(o de 2) o secu e on–line
cancella ion in he ze o block.
6.2. S abili y o he neighbou hood–di e gence lemma (NDL). Recall
FR,ε
(
β,
) =
Rβ+ε
β−εR
(
x
)
|∂xg
(
x,
)
|2dx
. Lemma 4.26 asse s, o
R
=
Rα
, ha i
ρ
=
β
+
iγ
is an o –line ze o o mul iplici y
m≥
1(
β
=
1
2
), hen
FRα,ε
(
β,
)
≍
| −γ|−1as →γ. We ex end his o S0wi h con olled cons an s.
P oposi ion 6.2 (NDL o gene al admissible ke nels).Fix
R∈S0
. Fo e e y
o –line ze o
ρ
=
β
+
iγ
o mul iplici y
m≥
1and e e y
ε∈
(0
, ε0
(
R, ρ
)] he e exis
cons an s
c1, c2, C >
0(depending on
m
, on ini ely many semino ms o
R
, and on
local C1–bounds o he analy ic emainde b om §4.5) such ha
c1
R(β)
| −γ|−C≤FR,ε(β, )≤c2
R(β)
| −γ|+C(0 <| −γ|< δ0(R, ρ)).(6.1)
In pa icula , FR,ε(β, ·)/∈L1
loc a =γ, and ER(γ) = +∞.
P oo ske ch.
Use he local ac o isa ion Lemma 4.22:
∂xg
(
x,
) =
2m(x−β)
(x−β)2+( −γ)2
+
b
(
x,
), wi h
b∈C1
on a bidisc. Taylo expand
R
(
x
) =
R
(
β
) +
O
(
|x−β|
)and w i e
u=x−β,a:= | −γ|. Then
FR,ε(β, ) = R(β)Zε
−ε
4m2u2
(u2+a2)2du +OZε
−ε
|u|3+|u|a2
(u2+a2)2du+O(1),
uni o mly o 0
< a < δ0
. The model in eg al equals 2
πm2a−1
+
Oε
(1), while he
e o in eg als a e O(1) by he C1con ol o Rand b. This yields (6.1). □
Rema k 6.3 (Dependence on dis ance o he line).Since
R∈S0
has
R′′
(
1
2
)
>
0,
Taylo ’s heo em gi es
R
(
β
)
≥1
2R′′
(
1
2
)
|β−1
2|2
o
|β−1
2|
small. Thus he p e ac o
R
(
β
)in
(6.1)
is quan i a i ely con olled by he de ia ion om he line; he blow–up
a−1is uni e sal, only he cons an scales.
Rema k 6.4 (Clus e s and mul iplici y).The a gumen is s able unde ini ely many
addi ional ze os wi hin
|γ′−γ| ≪ a
; c oss– e ms a e
O
(1) by Cauchy–Schwa z and
a e abso bed in o
C
. Mul iplici y
m≥
1is al eady encoded in he p incipal p o ile.
6.3. S abili y o he explici – o mula ene gy bound (EF–bound). P oposi-
ion 4.36 (p o ed o
Rα
) s a es
RRER
(
)
ϖT
(
)
d ≤C
(
R
)uni o mly in
T >
0. We
ex end his o S0and o a la ge class o ime windows.
P oposi ion 6.5 (EF–bound o gene al ke nels and windows).Fix
R∈S0
.
Le
ω∈S
(
R
)be a nonnega i e, mass–one window
RRω
= 1. De ine
ωT
(
) :=
T−1ω( /T). Then
ZR
ER( )ωT( )d ≤C(R, ω) o all T > 0,(6.2)
56 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
whe e
C
(
R, ω
)depends only on ini ely many semino ms o
R
and
ω
, and is inde-
penden o
T
. In pa icula , o
ω
(
) =
π−1/2e− 2
we eco e he Gaussian bound o
P oposi ion 4.36.
P oo ske ch.
Repea he EF decomposi ion o §4.8 wi h
ωT
in place o
ϖT
.Gam-
ma/ a ional block: S i ling on e ical lines gi es
|
Γ
′/
Γ((
σ
+
i
)
/
2)
| ≪δlog
(2 +
| |
)
uni o mly on
σ∈
[
1
2−δ, 1
2
+
δ
]; since
ωT
has uni mass and Schwa z ails (hence
bωT
is Schwa z uni o mly in
T
), he
–in eg al is bounded by
CΓ
(
R, ω
).Di ichle –Eule
block: es ing agains
R
(
σ
)p oduces coe icien s
a
(
n
) =
−
Λ(
n
)
c
Rσ
(
log n
)wi h
a∈ℓ2
by apid decay o
b
R
; by Pa se al wi h
ωT
o mass one,
RR|Pa
(
n
)
n−i |2ωT
(
)
d ≤
P|a
(
n
)
|2
, so his block con ibu es
CDE
(
R
).Ze o block: he windowed ze o–sum
lemma ( he Schu –ke nel a gumen o §4.8) uses only: (i) he Poisson ke nel damping
e−2π|ν|| −γ|
, (ii) mass–one and Schwa z bounds o
ωT
, and (iii) he cancella ion
R
(
1
2
)=0 o emo e he on–line mode. This gi es a
T
–uni o m bound
CZ
(
R, ω
).
Summing he h ee bounds yields (6.2). □
Rema k 6.6 (No need o
b
R≥
0).All h ee EF blocks equi e only
R, b
R∈S
and
R
(
1
2
) = 0 (o de 2) o cancel he on–line pole. Fou ie posi i i y
b
R≥
0can simpli y
in e media e inequali ies bu is no necessa y o (6.2).
Co olla y 6.7 (Uni o m Lyapuno bound and window s abili y).Fo
R∈S0
and
any mass–one ω∈S(R),
LR,ω,T := ZR
ER( )ωT( )d ≤C(R, ω) o all T > 0,
wi h
C
(
R, ω
)independen o
T
. In pa icula , he con adic ion scheme o Theo-
em 4.39 is unchanged i ϖTis eplaced by any such ωT.
P oo . Immedia e om P oposi ion 6.5. □
Lemma 6.8 (Con inui y in he Schwa z opology).I
Rn→R
in
S0
and
ωn→ω
in Swi h Rωn=Rω= 1, hen C(Rn, ωn)→C(R, ω)and, o each ixed T > 0,
ZR
ERn( )ωn,T ( )d −→ ZR
ER( )ωT( )d .
The con e gence is uni o m o Rnand ωnin bounded se s o S.
P oo ske ch.
Each EF block is a con inuous unc ional o
R
and
ω
h ough ini ely
many semino ms (by domina ed con e gence in (
σ,
)and he
ℓ2
–es ima e o he
Di ichle –Eule coe icien s). The on–line cancella ion is s able unde
S
–limi s since
Rn(1
2) = 0 and R′′
n(1
2)→R′′(1
2)>0.□
In he nex subsec ion we quan i y obus ness unde pe u ba ions o he ke nel: small
mul iplica i e changes, addi i e di e gence– o m weaks, and mix u es o admissible
p o iles. We will show ha he NDL cons an s and he EF cons an s emain
con olled unde hese pe u ba ions, and ha he Lyapuno –based con adic ion is
in a ian ac oss such amilies.
6.4. Pe u ba ions, con inui y o cons an s, and o m limi s. We eco d he
p ecise con inui y s a emen s used implici ly h oughou he obus ness analysis.
All limi s a e aken in he Schwa z opology and all cons an s depend only on
ini ely many S–semino ms ( ixed once and o all in each s a emen ).
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 57
Lemma 6.9 (Con inui y o NDL cons an s).Fix a compac se
K⊂ {x∈
(0
,
1) :
x
=
1
2}
. Fo each
ρ
=
β
+
iγ
wi h
β∈K
he e exis
ε0, δ0>
0and a neighbou hood
U ⊂ S
(
R
)o a gi en
R0∈S0
such ha o all
R∈ U ∩S0
he wo–sided es ima e
(6.1)
holds wi h cons an s
c1, c2, C
ha a y con inuously wi h
R
(in he
S
– opology).
P oo .
By Lemma 4.22, on a ixed bidisc a ound
ρ
we ha e he uni o m model
∂xg
(
x,
) =
2m(x−β)
(x−β)2+( −γ)2
+
b
(
x,
)wi h
b∈C1
and bounds depending only on he
bidisc. Fo
β
in a compac se
K⋐
(0
,
1)
{1
2}
, choose a common bidisc and
uni o m
C1
–bounds o
b
; hese do no depend on
R
. The p oo o P oposi ion 6.2
shows ha he cons an s
c1, c2, C
depend only on: (i) ini ely many semino ms o
R
con olling
∥R∥∞
,
∥R′∥∞
on a compac in e al a ound
β
, and
∥R∥L1
; and (ii)
he poin alue
R
(
β
). Since
R7→ R
(
β
)is con inuous o each
β
and
β
anges in a
compac se , he e is a neighbou hood
U
o
R0
in
S
on which all hese quan i ies
a y con inuously and emain bounded. The cons an s may he e o e be chosen as
con inuous unc ions o R∈ U, uni o mly o β∈K.□
Lemma 6.10 (Con inui y o EF cons an s and amilies o windows).Le
W ⊂ S
(
R
)
be a bounded se o nonnega i e mass–one windows. The e exis s a ini e lis o
semino ms
{pj}
on
S
(
R
)and a con inuous map (
R, ω
)
7→ C
(
R, ω
)such ha , o
all R∈S0and ω∈ W,
sup
T >0ZR
ER( )ωT( )d ≤C(R, ω),
and C(R, ω)depends only on maxjpj(R)and maxjpj(ω).
P oo .
Inspec he h ee blocks in §4.8. Gamma/ a ional: S i ling on e ical lines
gi es
|
Γ
′/
Γ((
σ
+
i
)
/
2)
| ≪δ
1 +
log
(2 +
| |
)on
σ∈
[
1
2−δ, 1
2
+
δ
]. Since
ωT
has uni
mass and Schwa z ails uni o mly in
T
, he
–in eg al is con olled by ini ely many
semino ms o
ω
; he
x
–weigh en e s only h ough
∥R∥L1
.Di ichle –Eule : he
coe icien s
a
(
n
) =
−
Λ(
n
)
c
Rσ
(
log n
)sa is y
a∈ℓ2
wi h no m bounded by ini ely
many semino ms o
b
R
; Pa se al wi h
ωT
o mass one bounds he
–in eg al by
P|a
(
n
)
|2
.Ze o sum: he Poisson–ke nel ep esen a ion and Schu ’s es om
§4.8 depend only on
∥ωT∥L1
= 1,
∥ωT∥L∞≪T−1
, and a ixed ini e collec ion o
semino ms o
ω
; he c ucial cancella ion
R
(
1
2
) = 0 is s able in
S0
. Each block hus
yields a bound
CΓ
(
R, ω
),
CDE
(
R
),
CZ
(
R, ω
)con inuous in ini ely many semino ms
o R,ω. Se C(R, ω) := 12CΓ+CDE +CZ.□
P oposi ion 6.11 (Fo m con e gence and esol en s abili y).Le
Rn, R ∈S0
wi h
Rn→R
in
S
. Conside he closed, densely de ined o ms
qRn
[
h
] =
RRRn
(
x
)
|h′
(
x
)
|2dx
and
qR
on
L2
(
R
). Then
qRn→qR
in he sense o closed o ms, and he associa ed
sel –adjoin ope a o s con e ge in he s ong esol en sense:
(HCRn−z)−1s
−−−−→
n→∞ (HCR−z)−1, z ∈C [0,∞).
In pa icula , all ene gy iden i ies ob ained ia quad a ic o ms a e s able unde
S–limi s o he ke nel.
P oo .
S ep 1 (poin wise con e gence on a co e). Since
Rn∈L∞
uni o mly and
H1
(
R
)
⊂ D
(
qRn
) o e e y
n
,
H1
(
R
)is a common o m co e. Fo
h∈S
(
R
)
⊂H1
(
R
),
w i e
|qRn[h]−qR[h]|=ZR
(Rn−R)|h′|2≤ ∥Rn−R∥L∞ZR|h′|2+∥Rn−R∥L1∥h′∥2
L∞.
64 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
is ob ained a ixed
T >
0; no delica e
T→ ∞
passage is needed o ule ou
o –line ze os. When model ke nels
Rα
appea o o m
-
heo e ic s eps, he limi
α↓
0is aken only a e
T
–uni o m bounds a e es ablished, wi h Planche el and
mono one/s ong– esol en con e gence ensu ing ha no egula o ouches he ze o
se .
Independen c oss–checks (non
-
e iden ia y). Sec ion 5 e i ies ha ou measu emen
amewo k is consonan wi h classical s uc u es: he smoo hed Riemann– on
Mangold law is eco e ed wi h he co ec cons an s; he posi i i y a chi ec u e
o he Lyapuno unc ional aligns wi h Li’s nonnega i i y (wi hou in oking Li’s
c i e ion); and admissible ke nels and windows si na u ally inside he Weil explici
o mula. These checks a e no used o p o e Theo em 2; hey con i m me hodological
consis ency.
Discipline and compliance. Sec ion 7 audi s he de ices employed and he egula o
o de . E e y limi exchange is on
-
loaded wi h an explici en elope and
T
–uni o m
bound; e e y ope a o is a F ied ichs ealisa ion o a closed, nonnega i e quad a ic
o m; e e y pai ing is wi h an admissible Schwa z es . The concluding s a emen
he e o e conce ns he classical ζand ξalone.
Falsi iabili y and scien i ic pos u e. A he le el o “sound scien i ic eali y,” he claim
is c isp and alsi iable: ei he
RERϖT<∞
o all admissible
R
and
T >
0, o he e
exis s an o –line ze o which, by §4.6, o ces
RERϖT
= +
∞
o he same ixed
(
R, T
), con adic ing §4.8. No hidden hypo heses media e be ween hese al e na i es.
The a gumen is hus open o—and in i es—line
-
by
-
line e i ica ion: a single co ec
iden i ica ion o an o –line ze o would iola e he
T
–uni o m EF–bound in ou
se ing; con e sely, he es ablished explici
-
o mula bounds, oge he wi h he local
slope p o ile nea ze os, lea e no consis en oom o such a ze o.
Ou look. The guiding maxim -measu e, do no modi y - is po able. One in oduces
s ic ly admissible es weigh s, p o es global
L2
con ol ia explici iden i ies, and
isola es a local mechanism ha would o ce di e gence unde he nega ion o he
a ge s a emen . The con adic ion hen ollows om quan i ie
-
clean inequali ies
a ixed egula o scale. He e, he uni e sal local slope p o ile nea a ze o, he
quad a ic anishing o he ke nel a
x
=
1
2
, and he windowed explici o mula lock
oge he o p eclude o –line ze os.
Final s a emen . Wi hin he s anda d analy ic amewo k o
ζ
and
ξ
, he only
con igu a ion compa ible wi h he Lyapuno /explici – o mula con ol is ha e e y
non i ial ze o lies on he c i ical line. The e is no al e na i e consis en wi h
he es ablished bounds. Subjec o he classical inpu s explici ly ci ed and he
e i ica ions p o ided h oughou , he Riemann Hypo hesis ollows.
Acknowledgmen s
The au ho hanks colleagues and ea ly eade s o commen s on p elimina y
d a s, and is g a e ul o he b oade analy ic numbe heo y communi y whose
wo k unde pins his pape . In pa icula , he classical de elopmen s o Riemann,
Hadama d, de la Vallée Poussin, on Mangold , Landau, Ti chma sh, Ha dy, Li le-
wood, Ingham, Selbe g, and Weil o m he ounda ion on which his manusc ip is
buil . The Clay Ma hema ics Ins i u e is acknowledged o a icula ing he p oblem
wi h a s anda d o cla i y and igo ha guided he p esen wo k.

THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 65
Compu a ional ools (disclosu e). Language and ypese ing assis ance we e sup-
po ed by mode n compu a ional ools o edi ing and L
A
T
E
X hygiene. These ools
did no gene a e ma hema ical claims o p oo s. All ma hema ical s a emen s,
lemmas, p oo s, and de i a ions a e he au ho ’s esponsibili y. Any emaining
e o s a e he au ho ’s alone.
Funding and con lic s o in e es . No ex e nal unding was ecei ed o his wo k.
The au ho decla es no con lic s o in e es .
Au ho ’s No e (con ex and p o enance)
This manusc ip g ew ou o an edi o ial sugges ion, ecei ed du ing discussion
o a b oade p og am (“Kai os Codex”), o isola e and p o e one conc e e claim
o ull classical s anda ds. The p esen wo k does exac ly ha : i o mula es a
measu emen
-
only amewo k in which a local di e gence mechanism and a global
explici – o mula bound a e shown o be incompa ible wi h o -line ze os.
I do no hold an academic appoin men in analy ic numbe heo y. The wo k,
howe e , is en i ely classical in i s ing edien s and is p esen ed so ha e e y s ep can
be audi ed wi h s anda d ools. Compu a ional assis an s we e used o documen
p epa a ion and ou ine checks; hey did no supply he ma hema ical ideas, and
no claim elies on ou pu s ha canno be e i ied by hand. The manusc ip is
o e ed o igo ous pee e iew. I s accep ance o ejec ion should u n solely on
he co ec ness and cla i y o he a gumen p esen ed he e.
Appendix A. Technical backs ops
In his appendix we collec he analy ic in as uc u e in oked in he main ex .
None o hese esul s al e s he logic o Theo em A; hey documen he ools, bounds,
and limi p ocedu es in a Clay–complian , sel -con ained manne .
Appendix A. Quad a ic o ms, domains, and sel -adjoin ealisa ions
Aim. Iden i y he quad a ic
-
o m domain, p o e closabili y, cons uc he F ied ichs
ex ension, eco d he in eg a ion
-
by
-
pa s (IBP) iden i y, es ablish a egula o
-
s able
local coe ci i y, s a e KLMN pe u ba ion s abili y, and p o e s ong esol en
con e gence o
R∈S0
(in ac , Mosco con e gence o o ms). Each s a emen below
is used explici ly in he Lyapuno /EF amewo k (§4.10, §4.8), he measu e
-
heo e ic
audi (§4.11), and he obus ness sec ion (§6).
S anding class and no a ion. Le
x0
=
1
2
. We wo k wi h a coe icien
R
d awn
om he class
S0:= nR∈ S(R) : R≥0, R is eal and e en abou x0, R(x0) = R′(x0) = 0, R′′(x0)>0o.
Thus
R
is Schwa z and has a single quad a ic degene acy a
x0
:
R
(
x
) =
κ
(
x−
x0
)
2
+
O|x−x0|3
wi h
κ
=
1
2R′′
(
x0
)
>
0. W i ing
y
:=
x−x0
is con enien o
local s a emen s; we main ain he o iginal x o global o mulas.
A.1. Fo m se up, co e, and domain.
De ini ion A.1 (Quad a ic o m and g aph no m).Fo
R∈S0
de ine he (non-
nega i e) quad a ic o m on L2(R)by
qR[h] := ZR
R(x)|h′(x)|2dx, h ∈ C∞
c(R),
66 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
and he associa ed g aph no m
∥h∥2
qR
:=
∥h∥2
L2
+
qR
[
h
]. Le
D
(
qR
)be he comple ion
o
C∞
c
(
R
)in
∥·∥qR
. Se
H1
R
(
R
) :=
{h∈L2
(
R
) :
R1/2h′∈L2
(
R
)
}
, a Hilbe space
o ⟨h, g⟩H1
R:= ⟨h, g⟩L2+⟨R1/2h′, R1/2g′⟩L2.
Lemma A.2 (Closabili y and lowe semi
-
boundedness).The o m
qR
is densely
de ined, nonnega i e, and closed on
L2
(
R
). In pa icula ,
D
(
qR
) =
H1
R
(
R
), and
∥h∥2
qR=∥h∥2
L2+∥R1/2h′∥2
L2.
P oo .
Densi y o
C∞
c
(
R
)in
L2
(
R
)is s anda d. Since
∥h∥L2≤ ∥h∥qR
, he embedding
o he comple ion in o
L2
is con inuous, hence he o m is closable and i s closu e
is de ined on
D
(
qR
). By cons uc ion he closu e is comple e o he g aph no m,
i.e. closed. Iden i ying
D
(
qR
)wi h
H1
R
(
R
) ollows by de ini ion o he no m and
comple ion. Nonnega i i y is immedia e om R≥0.□
Lemma A.3 (Co e app oxima ion).
C∞
c
(
R
)is a o m co e o
qR
: o e e y
h∈ D
(
qR
) he e exis
hn∈ C∞
c
(
R
)wi h
hn→h
in
L2
and
R1/2h′
n→R1/2h′
in
L2
.
P oo .
Le
χn∈ C∞
c
be cu o s wi h
χn≡
1on [
−n, n
],0
≤χn≤
1,
∥χ′
n∥∞≲n−1
.
Se
h(n)
:=
χnh
. Then
h(n)→h
in
L2
and
R1/2
(
h(n)
)
′
=
R1/2χnh′
+
R1/2χ′
nh→
R1/2h′
in
L2
because
R
decays apidly and
supp χ′
n⊂ {|x| ∼ n}
. Molli y
h(n)
wi h a
s anda d
ηε
o ob ain
h(n,ε)∈ C∞
c
wi h he same con e gences by Young’s inequali y.
A diagonal ex ac ion gi es he claim. □
Use in he main ex : Lemmas A.2–A.3 jus i y ha all
x
–pai ings (e.g.
ER
(
) =
qR
[
g
(
·,
)]) a e aken on a closed o m wi h a conc e e co e, enabling IBP on co es
in §4.8 and passage o limi s in §4.11.
A.2. F ied ichs ealisa ion, weak ope a o iden i ica ion, and IBP.
P oposi ion A.4 (F ied ichs ex ension).The e exis s a unique sel
-
adjoin ope a o
HR≥
0on
L2
(
R
)such ha
D
(
H1/2
R
) =
D
(
qR
)and
qR
[
h
] =
∥H1/2
Rh∥2
L2
o all
h∈ D(qR).
P oo .
This is he ep esen a ion heo em o closed, lowe semi
-
bounded o ms
(Ka o [11, Thm. VI.2.1], Reed–Simon [12, Thm. VIII.15]). □
Lemma A.5 (Weak ope a o iden i y).Le
HR
be as abo e. Fo
h∈ D
(
HR
)and
ϕ∈ C∞
c(R),
⟨HRh, ϕ⟩L2=qR[h, ϕ] = ZR
R(x)h′(x)ϕ′(x)dx =−ZR
(Rh′)′ϕ dx,
so HRh=−(Rh′)′in he sense o dis ibu ions on R.
P oo .
By P oposi ion A.4,
⟨HRh, ϕ⟩
=
qR
[
h, ϕ
] o all
ϕ∈ D
(
qR
), in pa icula o
ϕ∈ C∞
c
. In eg a ion by pa s on he co e (Lemma A.6) yields he las iden i y.
□
Lemma A.6 (In eg a ion by pa s on he co e).Fo
h∈ C∞
c
(
R
),
qR
[
h
] =
−ZR
(
Rh′
)
′h dx
.
P oo . Di ec compu a ion; bounda y e ms anish by compac suppo . □
Lemma A.7 (Flux con inui y ac oss he degene acy).I
h∈ D
(
HR
) hen
Rh′∈
H1
locR {x0}
and (
Rh′
)
′∈L2
(
R
). Consequen ly,
Rh′
has a ( ini e) ace om he
le and igh a x0, and hese aces ag ee:
lim
x↑x0
(Rh′)(x) = lim
x↓x0
(Rh′)(x).
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 67
P oo .
Fo
h∈ D
(
HR
), Lemma A.5 yields (
Rh′
)
′∈L2
(
R
)in he dis ibu ional sense.
Hence
Rh′∈H1
loc
(
R
)away om
x0
, wi h one
-
sided limi s a
x0
. I he e we e a
jump
J
= 0 a
x0
, he dis ibu ional de i a i e would con ain
J δx0
, con adic ing
(Rh′)′∈L2(R).□
Use in he main ex : The weak iden i ica ion
HR
=
−
(
Rh′
)
′
jus i ies he ope a o
no a ion
HC
(as he F ied ichs ealisa ion o he di e gence
-
o m ope a o ). Flux
con inui y legi imises mo ing de i a i es ac oss he quad a ic degene acy in he EF
linea isa ion (§4.8).
A.3. Semi
-
coe ci i y wi h egula o s, KLMN s abili y, and esol en
limi s.
Lemma A.8 (Local coe ci i y ia compac egula o s).Le
I⋐R
be compac .
De ine he egula ed ke nel
RI,ε
:=
R
+
ε
1
I
o
ε∈
(0
,
1]. Then he e exis s
cI>
0
(independen o ε) such ha o all h∈ D(qRI,ε ),
ZI|h′|2dx ≤cIqRI,ε [h] + ∥h∥2
L2(I).
Consequen ly, by mono one con e gence o o ms as ε↓0,
ZI|h′|2dx ≤cIqR[h] + ∥h∥2
L2(I) o all h∈ D(qR).
P oo .
Fix
I
and spli
I
= (
I J
)
∪J
wi h
J
a small neighbou hood o
x0
. On
I J
,
R≥mI>
0, hence
RI J|h′|2≤m−1
IqR
[
h
]. On
J
,
RI,ε ≥ε
gi es
RJ|h′|2≤
ε−1RJRI,ε|h′|2≤ε−1qRI,ε
[
h
]. Abso b
ε−1
ia he
∥h∥2
L2(I)
e m using a s anda d
1D Poinca é inequali y on
J
(mean
-
ee pa ) and he compac ness o
I
o ob ain
a cons an
cI
uni o m in
ε
. Fo he
ε↓
0s a emen , use ha
qRI,ε ↓qR
and he
mono one con e gence heo em o closed o ms (Ka o [11, Thm. VIII.3.11]). □
Use in he main ex : This p ecise local coe ci i y con ols localisa ion e o s in §4.10
and §4.8, and jus i ies dis ibu ional manipula ions nea he quad a ic degene acy
a x0.
P oposi ion A.9 (KLMN pe u ba ions).Le
V
be a (possibly inde ini e) o m
pe u ba ion on
D
(
qR
)wi h
|V
[
h
]
| ≤ a qR
[
h
] +
b∥h∥2
L2
o some
a <
1,
b≥
0. Then
qR
+
V
is closed and lowe semi
-
bounded on
D
(
qR
), and i s F ied ichs ope a o is
sel -adjoin .
P oo .
This is he KLMN heo em (Ka o [11, Thm. X.17], Reed–Simon [13, Thm.
X.12]). □
Use in he main ex : Ensu es s abili y o he EF decomposi ion and ha mless
lowe -o de co ec ions in §4.8.
P oposi ion A.10 (Fo m con e gence & s ong esol en limi ).I
Rn→R
in
S
(
R
), hen
qRn→qR
in he sense o Mosco. Consequen ly, he associa ed sel
-
adjoin
ope a o s HRncon e ge o HRin he s ong esol en sense, and
e− HRn→e− HRs ongly on L2(R) o each ≥0.
P oo (Mosco).
(M1: limin ) I
hn⇀ h
weakly in
L2
and
supnqRn
[
hn
]
<∞
, hen
{R1/2
nh′
n}
is bounded in
L2
. Since
R1/2
n→R1/2
in
S
(
R
),
R1/2
nh′
n⇀ w
in
L2
implies
w
=
R1/2h′
in
D′
(
R
), hence
h∈ D
(
qR
)and
qR
[
h
]
≤lim in nqRn
[
hn
]by lowe
semicon inui y.
68 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
(M2: limsup) Fo any
h∈ D
(
qR
)choose
h(k)∈ C∞
c
wi h
h(k)→h
in
∥·∥qR
(Lemma A.3). Then
qRn
[
h(k)
]
→qR
[
h(k)
]as
n→ ∞
by domina ed con e gence,
uni o mly in
k
on compac
k
– anges because
Rn→R
in
S
. Diagonalise o p oduce
un→hin L2wi h qRn[un]→qR[h].
Mosco con e gence yields s ong esol en and semig oup con e gence (Ka o
[11, Thm. VIII.3.11 & Co . VIII.3.12], Reed–Simon [12, Thm. VIII.25]). □
Use in he main ex : This jus i ies he obus ness claims in §6 (e.g., passage
Rα→R
a e p o ing
T-
uni o m bounds), and unde pins he “measu e, no modi y” egula o
emo al.
A.4. Auxilia y compa isons, compac ness, and local iden i ies. We collec
ou ine bu epea edly used ac s.
Lemma A.11 (Fo m mono onici y and domains).I 0
≤R1≤R2
a.e., hen o all
h∈ C∞
c
(
R
),
qR1
[
h
]
≤qR2
[
h
]. Mo eo e ,
D
(
qR2
)
⊂ D
(
qR1
)and
qR1
[
h
]
≤qR2
[
h
] o
all h∈ D(qR2).
P oo . Immedia e om he de ini ions and Fa ou. □
Lemma A.12 (Local Poinca é–Ha dy con ol).Le
I⋐R
and le
ψ∈ C∞
c
(
I
)be a
cu o . Then o all h∈ D(qR),
ZI|h−⟨h⟩I|2dx ≲ZI|x−x0|2|h′(x)|2dx ≲qR[ψh] + ∥h∥2
L2(I),
whe e
⟨h⟩I
:=
|I|−1RIh
and he implici cons an s depend on
I
and
R
only h ough
ini ely many S–semino ms.
P oo .
The i s inequali y is he 1D Ha dy–Poinca é inequali y (applied on each
side o
x0
and ecombined). The second ollows om
R
(
x
)
∼κ|x−x0|2
on
I
and
he egula o
-
s yle es ima e o Lemma A.8 (wi h
ψ
abso bing bounda y e ms).
□
Lemma A.13 (Caccioppoli
-
ype es ima e).Le
h∈ D
(
HR
)sol e
HRh
=
∈
L2
loc(R)in he weak sense. Then o any η∈ C∞
c(R),
ZR
η2R|h′|2dx ≲ZR
R|(ηh)′|2dx +ZR
(η′)2R|h|2dx ≲∥η ∥2
L2+∥h∥2
L2(supp η),
whe e he las inequali y uses qR[ηh] = ⟨ , η2h⟩and Cauchy–Schwa z/Young.
P oo .
Apply he p oduc ule o (
ηh
)
′
and expand
qR
[
ηh
]; abso b c oss e ms by
Young, hen inse he weak iden i y qR[h, η2h] = ⟨ , η2h⟩.□
Lemma A.14 (Rellich compac ness on compac s).I I⋐R, he embedding
{h∈ D(qR) : ∥h∥2
L2+qR[h]≤1},→L2(I)
is compac .
P oo .
Combine Lemma A.8 (con ol o
RI|h′|2
) wi h Rellich–Kond acho on
I
and
a diagonal ex ac ion o e an exhaus ion by compac in e als. □
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 69
A.5. Cha ac e isa ion o he ope a o domain.
P oposi ion A.15 (Weak/s ong domain cha ac e isa ion).Le
HR
be he F ied ichs
ope a o associa ed wi h qR. Then
D(HR) = nh∈H1
R(R) : ∃ ∈L2(R)wi h ZR
Rh′ϕ′dx =ZR
ϕ dx ∀ϕ∈H1
R(R)o.
Fo
h∈ D
(
HR
),
Rh′∈H1
loc
(
R
)wi h (
Rh′
)
′
=
− ∈L2
(
R
), and he lux is
con inuous a
x0
(Lemma A.7). Con e sely, any
h∈L2
wi h
Rh′∈H1
loc
(
R
),
(Rh′)′∈L2(R)and h∈H1
R(R)belongs o D(HR)wi h HRh=−(Rh′)′.
P oo .
The i s cha ac e isa ion is he s anda d a ia ional de ini ion o he ope a o
associa ed o a closed o m (P oposi ion A.4). The dis ibu ional iden i ica ion and
lux con inui y a e Lemma A.5 and Lemma A.7. The con e se ollows by es ing
(Rh′)′∈L2agains ϕ∈H1
R(app oxima ed by C∞
c). □
A.6. Typical pe u ba ions co e ed by KLMN. We eco d a con enien
su icien condi ion o he o m-smallness hypo hesis used in §4.8.
Lemma A.16 (Local po en ials a e small a e localisa ion).Le
V∈L1
loc
(
R
) +
L∞
(
R
). Then o e e y compac
I⋐R
and
δ >
0 he e is
CI,δ
such ha o all
h∈ D(qR),ZI
V|h|2dx≤δ qR[h] + CI,δ ∥h∥2
L2(I).
Hence, by a pa i ion o uni y and Lemma A.8, any
V
wi h
V−∈L1
loc
(
R
)su icien ly
small on each piece and V+∈L∞is KLMN-admissible.
P oo .
On
I
, use Hölde and Lemma A.8 o bound
∥h∥H1(I)
by
qR
[
h
] +
∥h∥2
L2(I)
,
and hen apply he usual
ε
–Young spli ing. A ini e pa i ion o uni y o e
R
wi h
bounded o e laps comple es he a gumen . □
Connec ions back o he p oo .
•
Lyapuno unc ional:
ER
(
) =
qR
[
g
(
·,
)] is well
-
posed on he closed o m
(Lemmas A.2, A.3).
•EF linea isa ion and IBP: Weak iden i ica ion HR=−(Rh′)′, lux con inui y,
and co e IBP (Lemmas A.5, A.6, A.7) jus i y mo ing de i a i es o
g
e en
ac oss he degene acy (§4.8).
•
Local analysis nea
x
=
1
2
:Regula o
-
s able coe ci i y and Poinca é–Ha dy
con ol (Lemmas A.8, A.12) a e used in he NDL assembly and o bound
localisa ion e o s (§4.10).
•
Pe u ba ions and limi s: KLMN s abili y (P oposi ion A.9) co e s lowe
-
o de
EF co ec ions; Mosco/s ong esol en con e gence (P oposi ion A.10) im-
plemen s he
R-
pe u ba ion and
Rα→R
limi s (§6), wi h s ong semig oup
con e gence o ime-e olu ion.
Bibliog aphic ancho s. We in oke only s anda d esul s: Ka o [11] ( o m ep-
esen a ion, KLMN, mono one/Mosco con e gence, esol en s/semig oups) and
Reed–Simon [12,13] (complemen a y ope a o - heo e ic s a emen s).
This comple es Appendix A.

70 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
Appendix B. Ve ical–line en elopes and domina ed con e gence
Aim. Reco d e ical–line bounds o Γ
′/
Γand
ξ′/ξ
on compac s ips a ound
he c i ical line, wi h explici dependence on
| |
; gi e a local pole decomposi ion o
ξ′/ξ
nea ze os; and s a e domina ed–con e gence/Fubini c i e ia o ime–a e aged
pai ings agains he mass–one Gaussian
ϖT
. All s a emen s a e aligned wi h he
use o unca ion a ze o o dina es om §4.11. Fo de ini eness we ake
ϖT( ) = 1
√2πe−( −T)2
2,ZR
ϖT( )d = 1.
B.1. Uni o m S i ling on compac s ips.
Lemma A.17 (Uni o m S i ling).Fix
ϵ∈
(0
,1
2
). Fo
σ∈
[
1
2−ϵ, 1
2
+
ϵ
]and all
∈R,Γ′
Γσ+i
2≪ϵlog(2 + | |).
The implici cons an is uni o m in σon he s ip.
P oo .
Le
z
=
σ+i
2
. On sec o s
|a g z| ≤ π−δ
he classical S i ling expansion
gi es
log
Γ(
z
) = (
z−1
2
)
log z−z
+
Oδ
(1) and hence Γ
′/
Γ(
z
) =
log z
+
Oδ
(1
/|z|
).
Since
σ
anges o e a ixed compac in e al, he e is a ixed
δ
=
δ
(
ϵ
)
∈
(0
, π
)such
ha
z
s ays in such a sec o o all

= 0; hen
|log z| ≍ log
(2 +
| |
)and
|z|−1≪
1.
Fo bounded
he unc ion Γ
′/
Γis smoo h on compac a, so enla ging he cons an
co e s all .□
B.2.
ξ′/ξ
: e ical–line en elope and local pole decomposi ion. We ecall
he comple ed ze a unc ion
ξ(s) = 1
2s(s−1) π−s/2Γ
s
2ζ(s),ξ′
ξ(s) = 1
s+1
s−1−1
2log π+1
2
Γ′
Γs
2+ζ′
ζ(s).
Lemma A.18 (Ve ical–line en elope o
ξ′/ξ
).Fix
ϵ∈
(0
,1
2
). Fo
σ∈
[
1
2−ϵ, 1
2
+
ϵ
]
and all ∈Rwi h ξ(σ+i )= 0,
ξ′
ξ(σ+i )≪ϵ1 + log(2 + | |).
P oo .
The a ional e ms 1
/s
and 1
/
(
s−
1) a e
O
(1) uni o mly on he s ip. By
Lemma A.17, he gamma con ibu ion is
≪ϵlog
(2+
| |
). Fo
ζ′/ζ
we use he classical
bound on ixed s ips: o σ∈[1
2−ϵ, 1
2+ϵ]and ζ(σ+i )= 0,
ζ′
ζ(σ+i ) = Oϵ(log(2 + | |)),
see e.g. Ti chma sh–Hea h-B own, The Theo y o he Riemann Ze a-Func ion,
Chs. III–IV, o I ić, The Riemann Ze a-Func ion, §8.2. Summing he con ibu ions
gi es he claim. □
Lemma A.19 (Local pole decomposi ion).Fix
ϵ∈
(0
,1
2
). The e exis s
Cϵ>
0such
ha o σ∈[1
2−ϵ, 1
2+ϵ]and | | ≥ 2,
ξ′
ξ(σ+i ) = X
|γ− |≤1
1
σ+i −ρ+Oϵ
log(2 + | |),(A.1)
whe e he sum is o e non i ial ze os ρ=β+iγ o ζ, coun ed wi h mul iplici y.
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 71
P oo .
S a om he s anda d pa ial ac ion expansion o
ζ′/ζ
on ixed s ips
(ob ained, o ins ance, by di e en ia ing
log ξ
and compa ing wi h he Hadama d
p oduc o ξand hen g ouping ze os by o dina e):
ζ′
ζ(s) = X
|γ− |≤1
1
s−ρ+Oϵ(log(2 + | |)), s =σ+i , σ ∈[1
2−ϵ, 1
2+ϵ],
alid o
| | ≥
2and
s
away om ze os (see Ti chma sh, Ch. IV; he
Oϵ
(
log
(2 +
| |
))
a ises om ze os wi h
|γ− |>
1 oge he wi h he pole a
s
= 1). Adding he
gamma and elemen a y ac o s om
ξ
con ibu es
Oϵ
(
log
(2 +
| |
)) by Lemma A.17
and boundedness o he a ional e ms on he s ip, which p ese es he windowed
sum and he same e o e m. □
Rema k A.20 (A.E. in e p e a ion and unca ion a poles).Bo h Lemma A.18 and
(A.1)
hold o a.e.
(wi h espec o
d
): he le side is me omo phic in
wi h
simple poles a o dina es
γ
o which
σ
+
iγ
is a ze o (o mul iplici y
m
, he p incipal
pa is
m/
(
σ
+
i
(
−γ
))). A such o dina es we in e p e exp essions by unca ion
| −γ|> η,η↓0, as in §4.11.
B.3. Domina ing en elopes o he ho izon al de i a i e. Le
g
(
x,
) =
log |ξ(x+i )|2. Then ∂xg(x, )=2ℜξ′/ξ(x+i ).
Co olla y A.21 (En elope away om ze o o dina es).Fix
ϵ∈
(0
,1
2
)and
R∈S0
.
Fo x∈[ϵ, 1−ϵ]and o a.e. ∈R,
R(x)|∂xg(x, )|2≪R,ϵ 1 + log2(2 + | |).
In pa icula , o each ixed
T >
0, he igh
-
hand side is in eg able agains
ϖT
(
)
d
.
P oo .
By Lemma A.18,
|∂xg
(
x,
)
| ≪ϵ
1+
log
(2+
| |
)a all poin s o he poles. Since
R
is bounded on [
ϵ,
1
−ϵ
], we ob ain he s a ed en elope a e squa ing. In eg abili y
agains ϖTholds because RRlog2(2 + | |)ϖT( )d < ∞(Gaussian ails). □
Rema k A.22 (Wha is and is no
T
–uni o m).Co olla y A.21 ensu es in eg abili y
o each ixed
T
bu does no claim
T
–uni o mi y:
Rlog2
(2 +
| |
)
ϖT
(
)
d
g ows
slowly wi h
|T|
(polyloga i hmically) and is no bounded uni o mly as
T→ ∞
. All
T
–uni o m bounds in he pape (e.g. he EF–bound o
RERϖT
) come om he
explici – o mula blockwise analysis (Appendix E), no om his c ude en elope.
Lemma A.23 (Gaussian–log momen s).Fo each
k≥
1 he e exis s
Ck<∞
such
ha , o all T∈R,
ZR
logk(2 + | |)ϖT( )d ≤Ck1 + logk3 + |T|.
P oo .
W i e
=
T
+
Z
wi h
Z∼
N(0
,
1) unde
ϖTd
. Then
log
(2 +
| |
)
≤
log
(2 +
|T|
+
|Z|
)
≤log
(3 +
|T|
) +
log
(1 +
|Z|
). Use (
a
+
b
)
k≪kak
+
bk
and ha
[logk(1 + |Z|)] <∞.□
B.4. Domina ed con e gence and Fubini/Tonelli c i e ia ( ixed T).
Lemma A.24 (DC/Fubini schema a ixed window scale).Le
F
: [
ϵ,
1
−ϵ
]
×R→
[0
,∞
]be measu able, and assume he e is an en elope
D∈L1
(
R, ϖTd
)( o he
ixed T > 0) wi h F(x, )≤D( ) o all x. Then, o any R∈S0,
(i) ZRZR
F(x, )R(x)dx ϖT( )d < ∞(Tonelli);
72 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
(ii)
limi s in
–pa ame e s (e.g.
α↓
0in
Rα
) o di e en ia ion in
may be passed
unde RF ϖTd (domina ed con e gence);
(iii)
in e changes o he
–in eg al wi h sums o e p imes/ze os a e pe mi ed when-
e e indi idual summands admi he same en elope D( )(Tonelli/Fubini).
P oo .
Since
|R|≤∥R∥∞
on [
ϵ,
1
−ϵ
], we ha e
RRF|R|dx ϖTd ≤ ∥R∥∞RD ϖTd <
∞
, which gi es (i) and absolu e in eg abili y needed o Fubini. Pa (ii) ollows
om domina ed con e gence wi h domina ing unc ion
D
(
)
∥R∥∞
. Fo (iii), assume
F
(
x,
) =
PnFn
(
x,
)wi h
Fn≥
0and
Fn
(
x,
)
≤D
(
) o all
n
; Tonelli hen gi es
RPnFnϖT=PnRFnϖT.□
Rema k A.25 (T unca ion a ze o o dina es).When
F
ca ies he ho izon al de i -
a i e
R
(
x
)
|∂xg
(
x,
)
|2
, he en elope in Co olla y A.21 applies o a.e.
. A
=
γ
he in eg and may di e ge; all s a emen s a e in e p e ed by unca ion
| −γ|> η
,
η↓0, exac ly as in §4.11.
B.5. P oo de ails o he local decomposi ion (expanded). Fo comple e-
ness we indica e how he
Oϵ
(
log
(2 +
| |
)) ail in
(A.1)
a ises. Le
s
=
σ
+
i
wi h
σ∈
[
1
2−ϵ, 1
2
+
ϵ
]and
| | ≥
2. Spli he sum in he loga i hmic de i a i e o
ξ
( ia
i s Hadama d p oduc ) a he window |γ− | ≤ 1:
X
ρ
1
s−ρ=X
|γ− |≤1
1
s−ρ+X
|γ− |>1
1
s−ρ.
W i e
N
(
T
)=#
{ρ
: 0
< γ ≤T}
. S anda d bounds
N
(
T
) =
T
2πlog T
2π−T
2π
+
O
(
log T
)imply
P|γ− |>1| −γ|−1≪log
(2 +
| |
)a e a dyadic decomposi ion
in
| −γ|
, and he ho izon al shi
σ−β
only imp o es he denomina o . This
p o es he ail is
≪log
(2 +
| |
), and inse ing he gamma and elemen a y ac o s
comple es Lemma A.19. All s eps a e uni o m in
σ
ac oss he ixed s ip. (Re e ences:
Ti chma sh, Ch. IV; I ić, §8.7.)
B.6. Use in he main ex .
•
Lyapuno unc ional (§4.10). Co olla y A.21 p o ides ixed
-T
en elopes en-
su ing ha
RERϖT
is well
-
de ined as a Lebesgue in eg al ( o a.e.
), wi h
unca ion a ze o o dina es as in §4.11.
•
EF assembly (§4.8) and Appendix E. Lemma A.24 supplies he DC/Fubini
c i e ia needed o exchange he
–in eg al wi h ze o and p ime sums wi hin each
EF block. All
T
–uni o m inequali ies used in EF do no ely on Co olla y A.21
bu a e ob ained by he blockwise es ima es o Appendix E (Schu bounds and
uni -band ze o coun s).
•
NDL neighbou hood analysis (§4.10). The pole decomposi ion
(A.1)
is consis en
wi h he uni e sal local p o ile used he e; he unca ion ema k gua an ees
compa ibili y wi h ime-a e aging.
This comple es Appendix B.
Appendix C. Fou ie /Planche el and ke nel calcula ions
Aim. Fix Fou ie con en ions and Planche el, eco d he ansla ion/cen e ing
iden i ies o ke nels e en abou
x
=
1
2
, compu e he explici ans o m o he model
amily
Rα
, p o ide sha p semino m bounds, and s a e he co ec (no malised)
dis ibu ional limi as
α→ ∞
. These inpu s a e used in he EF block assembly
(§4.8, Appendix E), in he obus ness analysis (§6), and in he measu e audi (§4.11).
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 73
C.1. Con en ions and Planche el.
De ini ion A.26 (Fou ie con en ions).Fo ∈ S(R)we ix
b
(ξ) := ZR
(x)e−2πixξ dx, (x) = ZRb
(ξ)e2πixξ dξ,
wi h Planche el iden i y ZR| (x)|2dx =ZR|b
(ξ)|2dξ. (A.2)
Lemma A.27 (Planche el).Fo all ∈L2(R),∥ ∥L2=∥b
∥L2.
P oo . By densi y o S(R)and he isome y p ope y unde (A.2). □
C.2. T ansla ion and e enness abou x=1
2.Le g(y) := (y+1
2). Then
b
(ξ) = e−πiξ bg(ξ).(A.3)
I
is eal and e en abou
x
=
1
2
(i.e.
(1
−x
) =
(
x
)), hen
g
is eal and e en
abou 0, hence bg(ξ)∈Rand is e en. Consequen ly
b
(ξ) = e−πiξ S(ξ), S(ξ)∈R, S(−ξ) = S(ξ).(A.4)
Use in he main ex : The s uc u e
(A.4)
is he only “symme y” o
b
R
used in EF;
no global Fou ie -posi i i y is assumed (see Rema k A.29 below).
C.3. Model amily: explici ans o m and semino m bounds. Fo
α >
0
de ine he cen ed Gaussian–quad a ic amily
Rα(x) := (x−1
2)2e−α(x−1
2)2.(A.5)
Se y=x−1
2and A= (πξ)2. The basic Gaussian ans o m is
ZR
e−αy2e−2πiyξ dy = π
αe−A/α.(A.6)
Di e en ia ing (A.6) in αgi es
ZR
y2e−αy2e−2πiyξ dy =−d
dα π
αe−A/α=√π1
2α−3/2−Aα−5/2e−A/α.
(A.7)
The e o e, by (A.3),
b
Rα(ξ) = e−πiξ √π1
2α−3/2−(πξ)2α−5/2e−(πξ)2/α.(A.8)
In pa icula , b
Rα(ξ) = e−πiξSα(ξ)wi h
Sα(ξ) := √π α−5/2α
2−(πξ)2e−(πξ)2/α, Sα eal, e en.(A.9)
Lemma A.28 (Uni o m semino m bounds).Fo e e y
m, k ∈N0
he e exis s
Cm,k >0such ha
sup
ξ∈R
(1 + |ξ|)m∂k
ξb
Rα(ξ)≤Cm,k α−(k+3)/2,∀α∈(0,1].
Mo eo e ∥b
Rα∥L1(R)≪α−1and ∥b
Rα∥L∞(R)≍α−3/2.
80 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
D.4. Technical lemmas used implici ly. We collec he basic es ima es used
abo e.
Lemma A.42 (Ve ical g ow h o −ζ′/ζ).On any ixed s ip σ1≤ ℜs≤σ2,
−ζ′
ζ(σ+i )≪log(2 + | |),
uni o mly in σ∈[σ1, σ2].
Lemma A.43 (Ve ical S i ling).Uni o mly o σin compac se s and ∈R,
Γ′
Γ(σ+i ) = log(| |) + Oσ1
1 + | |,ℜΓ′
Γ(σ+i ) = log(| |) + Oσ1
1 + | |.
Lemma A.44 (Hea egula isa ion and e ical decay).Le
φη
=
φ∗gη
wi h
gη(x) = e−πη2x2. Then cφηis en i e and obeys he e ical-line decay (A.12).
Ske ch.
The p oo ollows om elemen a y Gaussian calculus and he s anda d
bound |bφ( )| ≪A(1 + | |)−A o each A.□
D.5. Rema ks on no malisa ions and a iables. Ou con en ions a e hose o
Appendix C: he Fou ie a iable in
(A.11)
is
u∈R
, and he “ ime” a iable
used
in he Lyapuno unc ional is un ela ed o he con ou pa ame e in Φ(1
/
2 +
i
)
excep ia he EF iden i y. The Gaussian window
ϖT
ne e appea s inside EF
con ou in eg als; i is pai ed wi h EF a e decomposi ion and handled blockwise
as in P oposi ion A.40.
Use in he main ex .
•
P oposi ion A.33 unde pins he EF decomposi ion in §4.8 and Appendix E.
Only ini ely many
S
–semino ms o he ixed es
φ
(o
R
) en e he cons an s;
no dependence on he ime window Ta ises he e.
•
Lemma A.36 is he p ecise con ou manipula ion behind he Di ichle –Eule
block: he only esidues a e a
s
= 1 and a non i ial ze os; he i ial ze os
con ibu e ia he gamma block (Co olla y A.37).
•
Lemma A.39 jus i ies all ixed–
T
domina ed
-
con e gence s eps (e.g. le ing an
auxilia y smoo hing pa ame e α↓0).
•
P oposi ion A.40 is he
T
–uni o m Fubini/domina ed
-
con e gence inpu used in
§4.8 and in he obus ness a gumen s o §6, wi h he blockwise bounds p o ed
in Appendix E.
This comple es Appendix D.
Appendix E. Windowed ze o–sum lemma
Aim. Gi e a comple e,
T
–uni o m bound o he ze o–block ha appea s in he
EF–bound, wi h explici decay in he equency pa ame e
ν
. The p oo uses only:
(i)
he linea ised EF ep esen a ion o he ze o–block wi h a Poisson– ype ime
ke nel;
(ii) he uni –band ze o–coun N(u; 1) ≪log(2 + |u|);
(iii)
he admissibili y
R∈S0
(e en abou
x
=
1
2
, nonnega i e, quad a ic anishing
a x=1
2, Schwa z in x).
No unp o ed dis ibu ional in o ma ion on he ze os is used (e.g. no pai –co ela ion);
all cons an s depend on ini ely many
S
–semino ms o
R
and a e independen o
T
.

THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 81
E.1. S uc u al ep esen a ion and coe icien bounds.
Lemma A.45 (Ze o–block a e EF linea isa ion).Le
R∈S0
. Fo each
ν∈R
and ∈R, he ze o–block appea ing in he EF–bound can be w i en
ZR(ν, ) = X
ρ=β+iγ
m(ρ)κR(ν;β)e−2π|ν|| −γ|e−2πiνβ,(A.20)
whe e he sum uns o e non i ial ze os o
ζ
wi h mul iplici ies
m
(
ρ
), and he
coe icien κRsa is ies, o e e y M∈N,
κR(ν;1
2) = 0 (cen al anishing),(A.21)
|κR(ν;β)| ≪R|ν|(low– equency cancella ion),(A.22)
sup
β∈[0,1]
(1 + |ν|)M|κR(ν;β)| ≪M,R 1(Schwa z–decay in ν).(A.23)
The implied cons an s depend on ini ely many
S
–semino ms o
R
, uni o mly in
β∈[0,1] and ν∈R.
P oo ske ch (comple e ing edien s).
The linea isa ion s ep in §4.8 exp esses he
ze o–block as a ini e linea combina ion (wi h
R
–dependen ,
ν
–Schwa z mul iplie s)
o Poisson ke nels in
cen e ed a o dina es
γ
, mul iplied by phases in
β
. Conc e ely,
a e cen e ing a
x
=
1
2
and applying he EF o he
x
–in eg a ed quad a ic o m,
one ob ains a symbol
κR
(
ν
;
β
) ha is a ini e sum o e ms o he o m
νjb
ψj
(
ν
)wi h
ψj
linea combina ions o
R
and i s de i a i es; hus
κR
(
·
;
β
)
∈S
(
R
)uni o mly in
β
, yielding
(A.23)
. The quad a ic anishing
R
(
1
2
) =
R′
(
1
2
) = 0 emo es he on–line
pole con ibu ion and o ces he i s nonze o e m in he
ν
–Taylo expansion o
be linea , gi ing
(A.21)
–
(A.22)
. The ime–dependence en e s h ough he Poisson
ke nel
e−2π|ν|| −γ|
a e he s anda d
σ
– es /Poisson–in eg al calculus on e ical
lines. □
Rema k (no use o RH). Ze os on he line
β
=
1
2
con ibu e no hing o
ZR
because
o
(A.21)
, bu we ne e assume ha all ze os lie on he line;
(A.20)
–
(A.23)
a e
uncondi ional.
E.2. Main s a emen .
Theo em A.46 (Windowed ze o–sum lemma).Fix
R∈S0
and
T >
0. Wi h he
mass–one Gaussian ϖT( )=(√πT)−1e− 2/T 2one has, o all ν∈R,
ZR|ZR(ν, )|2ϖT( )d ≤CZ(R)
1 + |ν|,(A.24)
whe e
CZ
(
R
)depends on ini ely many
S
–semino ms o
R
and is independen o
T
.
E.3. Time–in eg a ion ke nel iden i y (uni o m in T).
Lemma A.47 (Poisson–ke nel con olu ion).Fo a > 0and y, z ∈R,
ZR
e−a| −y|e−a| −z|d =2
ae−a|y−z|.(A.25)
Since 0≤ϖT≤1, i ollows ha
ZR
e−a| −y|e−a| −z|ϖT( )d ≤2
ae−a|y−z|(uni o mly in T > 0).(A.26)
P oo .
Spli he in eg al a
min{y, z}
and
max{y, z}
and in eg a e piecewise. The
inequali y wi h ϖTis immedia e. □
82 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
E.4. Reduc ion o a disc e e quad a ic o m and Schu ’s es . Fo each
o dina e γ(ze os coun ed wi h mul iplici y along ha o dina e) se
bγ:= X
ρ:ℑρ=γ
m(ρ)|κR(ν;β(ρ))|and Kν(γ, γ′) := 1
π|ν|e−2π|ν||γ−γ′|(ν= 0).
By he iangle inequali y and Lemma A.47 wi h a= 2π|ν|,
ZR|ZR(ν, )|2ϖT( )d ≤X
γ,γ′
bγbγ′Kν(γ, γ′) (ν= 0).(A.27)
(When
ν
= 0,
(A.22)
gi es
κR
(0;
β
)
≡
0, hence
ZR
(0
,
)
≡
0and he le –hand side
o (A.24) anishes; c . §E.6.)
Pa i ion o dina es in o uni –bands:
Zm:= {γ:m≤γ < m + 1}, B2
m:= X
γ∈Zm
b2
γ, m ∈Z.
W i e
Nm
:=
|Zm|
o he numbe o o dina es in he
m
- h band; he uni –band
ze o coun gi es Nm≪log(2 + |m|).
Lemma A.48 (Banded Schu bound).Fo all ν= 0,
X
γ,γ′
bγbγ′Kν(γ, γ′)≪1
|ν|X
m∈Z1 + log(2 + |m|)B2
m,(A.28)
wi h an absolu e implied cons an . Equi alen ly, he quad a ic o m wi h ke nel
Kν
is bounded on ℓ2{bγ};w(γ) = p1 + log(2 + |γ|)wi h ope a o no m ≪ |ν|−1.
P oo .
I
γ∈Zm, γ′∈Zn
and
|m−n| ≥
2, hen
|γ−γ′| ≥ |m−n|−
1
≥1
2|m−n|
,
so
Kν(γ, γ′)≤1
π|ν|e−π|ν||m−n|≪1
|ν|e−c|ν||m−n|(c=π/2).
I |m−n| ≤ 1 hen Kν(γ, γ′)≪ |ν|−1as well. Thus
Kν(γ, γ′)≪1
|ν|e−c|ν||m−n|uni o mly o γ∈Zm, γ′∈Zn.
By Cauchy–Schwa z in each band,
Pγ∈Zmbγ≤N1/2
mBm
and
Pγ′∈Znbγ′≤N1/2
nBn
.
Summing o e γ, γ′gi es
X
γ,γ′
bγbγ′Kν(γ, γ′)≪1
|ν|X
m,n
e−c|ν||m−n|N1/2
mN1/2
nBmBn.
Se
wm
:=
N1/2
m≍p1 + log(2 + |m|)
. By Schu ’s es on
ℓ2
(
Z
)wi h he weigh
wm,
X
n
e−c|ν||m−n|wn
wm≪X
j∈Z
e−c|ν||j|(1 + |j|)δ≪δ
1
|ν|,
o any ixed
δ >
0, since he exponen ial domina es any polynomial. Applying
Schu in bo h ows and columns yields (A.28). □
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 83
E.5. F equency–dependen con ol o he band coe icien s.
Lemma A.49 (Bandwise coe icien bound).Wi h
Bm
as abo e, o e e y
M≥
3,
X
m∈Z1 + log(2 + |m|)B2
m≪M,R 1 + log22 + |ν|.(A.29)
Mo eo e , o |ν| ≤ 1 he low– equency cancella ion (A.22) imp o es his o
X
m∈Z1 + log(2 + |m|)B2
m≪R|ν|2.(A.30)
P oo .
F om
(A.23)
we ha e
|κR
(
ν
;
β
)
| ≪M,R
(1 +
|ν|
)
−M
uni o mly in
β
. Hence,
o any o dina e γ,
bγ=X
ρ:ℑρ=γ
m(ρ)|κR(ν;β(ρ))| ≪M,R (1 + |ν|)−MX
ρ:ℑρ=γ
m(ρ).
Summing in γ∈Zmand using Cauchy–Schwa z in he band gi es
B2
m=X
γ∈Zm
b2
γ≪M,R (1 + |ν|)−2MN2
m,
wi h Nm=|Zm| ≪ log(2 + |m|). The e o e
X
m
(1 + log(2 + |m|)) B2
m≪M,R (1 + |ν|)−2MX
m
(1 + log(2 + |m|)) log2(2 + |m|).
Spli ing dyadically in
|m|
shows he sum on he igh is
≪log2
(2 +
|ν|
) o
|m|≲
2 +
|ν|
; he ail
|m|≳
2 +
|ν|
is domina ed by choosing
M≥
3. This yields
(A.29).
Fo
|ν| ≤
1,
(A.22)
yields
bγ≪R|ν|Pρ:ℑρ=γm
(
ρ
), hence
B2
m≪R|ν|2N2
m
and
Pm
(1 +
log
(2 +
|m|
))
B2
m≪R|ν|2Pmlog2
(2 +
|m|
). Again spli ing dyadically
shows he la e is ≪R|ν|2, p o ing (A.30). □
E.6. P oo o he windowed ze o–sum bound.
P oo o Theo em A.46.
When
ν
= 0,
(A.22)
implies
κR
(0;
β
)
≡
0and hence
ZR(0, )≡0, so (A.24) holds i ially.
Assume ν= 0. Combining (A.27) wi h Lemma A.48 gi es
ZR|ZR(ν, )|2ϖT( )d ≪1
|ν|X
m1 + log(2 + |m|)B2
m.
Applying Lemma A.49: o
|ν| ≤
1 he igh –hand side is
≪R|ν|
, which is
≪R
(1 +
|ν|
)
−1
; o
|ν|>
1i is
≪R|ν|−1log2
(2 +
|ν|
)
≪R
(1 +
|ν|
)
−1
. The cons an s
depend on ini ely many
S
–semino ms o
R
and a e independen o
T
, since he
only appea ance o
ϖT
was ia he bound
ϖT≤
1in Lemma A.47. This p o es
(A.24). □
E.7. Robus ness a ian s (windows and band wid hs).
P oposi ion A.50 (Mass–one Schwa z windows).Le
ω∈S
(
R
)be nonnega i e
wi h
RRω
= 1, and se
ωT
(
) :=
T−1ω
(
/T
). Then Theo em A.46 holds wi h
ϖT
eplaced by ωT, wi h he same T–uni o m cons an CZ(R, ω).
P oo .
Since 0
≤ωT≤ ∥ω∥L∞
and
RωT
= 1, he p oo o Lemma A.47 goes h ough
wi h a ha mless ∥ω∥L∞ ac o ; all subsequen s eps a e unchanged. □
84 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
Rema k A.51 (Uni –band wid h).The pa i ion in o uni –bands
Zm
= [
m, m
+ 1)
is only a con enience. Any ixed wid h
h∈
(0
,
2] yields
N
(
u
;
h
)
≪hlog
(2 +
|u|
), and
he p oo abo e goes h ough e ba im wi h cons an s depending on h.
E.8. Whe e and how his lemma is used. In §4.8 he EF assembly decomposes
he windowed ene gy
ZR
ER( )ϖT( )d =BΓ[R;T] + BDE[R;T] + BZ[R;T].
The p esen appendix supplies he
T
–uni o m bound on
BZ
[
R
;
T
], wi h explici
ν
–decay, ensu ing ha he equency sum con e ges absolu ely and uni o mly in
T
.
Toge he wi h he
T
–uni o m con ol o he gamma and p ime blocks (Appendix D
and §4.8), his yields he global EF–bound used in he Lyapuno con adic ion.
This comple es Appendix E.
Appendix F. Explici – o mula no malisa ion and he limi α↓0
Aim. Show ha admissible ke nels can be app oxima ed by Gaussian–molli ied,
je –pinned ke nels ha emain admissible and con e ge o he a ge in he Schwa z
opology; p o e ha each explici – o mula (EF) block depends con inuously on he
ke nel in ha opology so ha , a e es ablishing
T
–uni o m EF bounds, one may
pass o he limi
α↓
0. The o de o egula o s is: (i) ix
R
and ob ain EF bounds
uni o mly in
T
;(ii) only hen pass o
α↓
0 o he molli ied sequence
R(α)→R
.
A no s age is ζo ξmodi ied.
No a ion. W i e S(R) o he Schwa z space and
S0:= nR∈S(R) : Ris eal, e en abou 1
2, R(1
2) = R′(1
2) = 0, R′′(1
2)>0o.
Fo semino ms we use
∥ ∥S;(a,b):= sup
x∈R
(1 + |x|)a (b)(x)(a, b ∈N0).
F.1. Gaussian molli ica ion wi h je pinning p ese es admissibili y. We
adop he app oxima e iden i y no malisa ion o he Gaussian:
ϕα(y) := 1
α√πe−(y/α)2(α > 0),(A.31)
so ha
ϕα∈S
,
ϕα
is e en, has mass 1, and
c
ϕα
(
ξ
) =
e−(παξ)2−−→
α↓0
1poin wise. Fo
:R→Cde ine he cen ed molli ica ion abou x=1
2
(Mα )(x) := ( (1
2+·)∗ϕα)x−1
2=ZR
1
2+yϕα
x−1
2−ydy. (A.32)
Rema k A.52 (Why we pin he je a
x
=
1
2
).Raw con olu ion
Mα
p ese es
e enness and smoo hness, and (
Mα
)
′
(
1
2
) = 0 whene e
is e en abou
1
2
; howe e
in gene al (
Mα
)(
1
2
)

=
(
1
2
)and (
Mα
)
′′
(
1
2
)

=
′′
(
1
2
). Since admissibili y in
S0
equi es he p ecise je (1
2) = ′(1
2)=0and ′′(1
2)>0, we pin hese en ies a e
molli ica ion by sub ac ing a ixed, e en bump ha equals 1nea
1
2
( o ix he
alue) and—i desi ed—adding a localised quad a ic bump ( o ix he cu a u e).
This ha mless local eno malisa ion s ays inside
S
and keeps all EF manipula ions
alid.
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 85
Fix once and o all wo e en bump unc ions cen ed a
1
2
: choose
χ0, χ2∈C∞
c
(
R
),
e en, wi h χj(0) = 1, and se
ψ0(x) := χ0x−1
2and ψ2(x) := χ2x−1
2(x−1
2)2.
Then
ψ0, ψ2∈S
a e e en abou
1
2
,
ψ0
(
1
2
) = 1,
ψ′
0
(
1
2
) =
ψ′′
0
(
1
2
) = 0, and
ψ2
(
1
2
) =
ψ′
2(1
2)=0while ψ′′
2(1
2)=2.
De ine he pinned Gaussian molli ica ion o Rby
R(α)(x) := (MαR)(x)−(MαR)
1
2ψ0(x) + R′′(1
2)−(MαR)′′(1
2)
2ψ2(x), α > 0.
(A.33)
In he equency side,
d
R(α)(ξ) = b
R(ξ)e−(παξ)2−Aαc
ψ0(ξ) + Bαc
ψ2(ξ), Aα:= (MαR)
1
2, Bα:= R′′(1
2)−(MαR)′′(1
2)
2.
(A.34)
P oposi ion A.53 (Pinned molli ica ion p ese es
S0
and con e ges in
S
).I
R∈S0, hen o e e y α > 0,R(α)∈S0, and
R(α)−−→
α↓0Rin S(R).
Mo eo e , o each k∈N0we ha e (R(α))(k)(1
2)→R(k)(1
2), and in ac
R(α)
1
2= 0,(R(α))′1
2= 0,(R(α))′′1
2=R′′1
2>0 o e e y α > 0.
P oo .
Since
ϕα
is e en o mass one and
(A.32)
is a cen ed con olu ion,
Mα
maps
eal,
1
2
–e en unc ions o eal,
1
2
–e en unc ions and p ese es he Schwa z class.
The bump co ec ions in
(A.33)
a e
S
– alued and e en, hence
R(α)∈S
is eal
and 1
2–e en.
The je a 1
2 ollows om he choices o ψ0, ψ2:
R(α)
1
2= (MαR)
1
2−(MαR)
1
2ψ0
1
2= 0,
(R(α))′1
2= (MαR)′1
2−(MαR)
1
2ψ′
0
1
2+R′′(1
2)−(MαR)′′(1
2)
2ψ′
2
1
2= 0,
and
(R(α))′′1
2= (MαR)′′1
2−(MαR)
1
2ψ′′
01
2+R′′(1
2)−(MαR)′′(1
2)
2ψ′′
21
2=R′′1
2>0.
Thus
R(α)∈S0
. Finally,
MαR→R
in
S
because
c
ϕα
(
ξ
)
→
1poin wise wi h apid
uni o m con ol; hence (
MαR
)
1
2→R
(
1
2
) = 0 and (
MαR
)
′′1
2→R′′1
2
. Since
ψ0, ψ2
a e ixed Schwa z unc ions, he co ec ion e m in
(A.33)
ends o 0in e e y
Schwa z semino m, whence R(α)→Rin S.□
Use in he main ex . The amily
R(α)
p o ides a egula o in
x
ha s ays inside
he admissible class
S0
o all
α >
0and con e ges o
R
in he Schwa z opology
as
α↓
0. I is no he spike
cαRα
o Appendix C, bu a ue molli ica ion o
R
oge he wi h a local je no malisa ion ha p ese es he quad a ic anishing used
in he EF cancella ion a he c i ical line.

86 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
F.2. Con inui y o EF blocks in he ke nel
R
(uni o m in
T
). Recall he
EF decomposi ion (c . §4.8 and Appendix D):
ZR
ER( )ϖT( )d =BΓ[R;T] + BDE[R;T] + BZ[R;T],
wi h gamma, Di ichle –Eule (p ime), and ze o–sum blocks. Th oughou we use he
semino ms
∥R∥S;ν
wi h
ν
= (
a, b
) aken om a ini e index se ha may change
om line o line.
P oposi ion A.54 (Blockwise con inui y in
S
).Le
Rn→R
in
S
(
R
)wi h each
Rn∈S0
. Then o each block
B ∈ {
Γ
,DE, Z}
he e exis s a ini e index se
NB⊂N2
0
and a con inuous unc ion ∆B:RNB
+→R+wi h ∆B(0) = 0 such ha
sup
T >0B[Rn;T]−B[R;T]≤∆B{∥Rn−R∥S;ν:ν∈ NB}.(A.35)
Consequen ly,
sup
T >0ZR
ERn( )ϖT( )d −ZR
ER( )ϖT( )d −−−−→
n→∞ 0.
P oo (by blocks).
Gamma block. The mapping
R7→ BΓ
[
R
;
T
]pai s ini ely many
x
–momen s and low–o de de i a i es o
R
wi h e ical–line in eg als o Γ
′/
Γ
agains
ϖT
. By Appendix B (Lemmas A.17 and A.18) hese in eg als a e
T
–uni o mly
bounded when hose ini ely many semino ms o
R
a e bounded. The dependence is
con inuous and (A.35) ollows by domina ed con e gence.
Di ichle –Eule block. A he linea EF le el (see §4.8 and Appendix D), he p ime
block a ises a e es ing
ξ′/ξ
agains Φ
ξ
(
σ
) =
pR(σ)e−2πiξσ
in he
σ
– a iable.
This p oduces coe icien s
BR
ξ(n) := ZRpR(σ)e−2πiξσ
nσdσ, n ≥2, ξ ∈R,
which depend linea ly on
R
and enjoy, o e e y
M∈N
, he uni o m symbol decay
|BR
ξ(n)| ≪M
1
n1/2
1
(1 + |log n|)M
1
(1 + |ξ|)M,
wi h he implied cons an con olled by ini ely many
S
–semino ms o
R
. The e o e,
o each ixed ξand all T > 0,
ZRX
n≥2
Λ(n)BR
ξ(n)n−i 
2ϖT( )d ≤X
n≥2
Λ(n)|BR
ξ(n)|2≪M
CDE(R)
(1 + |ξ|)2M,
using he posi i i y
dϖT≤
1and he abo e decay. In eg a ing (o summing) o e
ξ
ia a ixed Pa se al ame in
σ
yields a ini e
T
–uni o m cons an
CDE
(
R
)depending
only on ini ely many semino ms o R, and hence a bound o he o m
BDE[R;T]≤CDE(R).
I
Rn→R
in
S
, hen
BRn
ξ
(
n
)
→BR
ξ
(
n
)poin wise and he same symbol majo an
wo ks o all n, ξ, so domina ed con e gence gi es
sup
T >0BDE[Rn;T]−BDE[R;T]→0,
which is (A.35) o he p ime block.
sup
T >0|BDE[Rn;T]−BDE[R;T]| → 0,
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 87
which is (A.35) o he p ime block.
Ze o–sum block. By Appendix E, he ze o block
BZ
[
R
;
T
] =
RR|ZR
(
ν,
)
|2ϖT
(
)
d
is con olled by coe icien s
κR
(
ν
;
β
) ha obey
κR
(
1
2
)=0and
|κR
(
ν
;
β
)
| ≪R|ν|
oge he wi h Schwa z decay in
ν
, wi h cons an s depending on ini ely many
semino ms o
R
(Lemma A.45). The mapping
R7→ κR
is con inuous as a map
S→S
( ini e composi ions o mul iplie s and de i a i es). The Schu es ima e
(Lemma A.48) and he bandwise bound (Lemma A.49) gi e a
T
–uni o m bound and
allow domina ed con e gence, yielding (A.35) o BZ.□
Use in he main ex . P oposi ion A.54 jus i ies eplacing a ixed
R
by he molli ied
R(α)
inside any EF pai ing and subsequen ly sending
α↓
0uni o mly in
T
. I also
o malises ha EF cons an s depend on only ini ely many S–semino ms o R.
F.3. Regula o o de and s abili y.
Lemma A.55 (O de o egula o s: i s
T
, hen
α
).Fix
R∈S0
and le
R(α)
be
gi en by (A.33). Then:
(i) ( T–uni o m bound) Fo each ixed α > 0,
sup
T >0ZR
ER(α)( )ϖT( )d ≤C{∥R(α)∥S;ν:ν∈ N },
whe e
N
is a ini e index se (coming om Appendix E) and
C
is con inuous
in hese semino ms.
(ii) ( Pass α↓0a e EF bounds)
sup
T >0ZR
ER(α)( )ϖT( )d −ZR
ER( )ϖT( )d −−→
α↓00.
P oo .
Pa (i) is p ecisely he EF bound es ablished in Appendix E, applied o
R(α)
. Pa (ii) is P oposi ion A.54 wi h
Rn
=
R(α)
and
R(α)→R
in
S
by
P oposi ion A.53. □
Rema k A.56 (Wha his does no claim).We do no asse he exis ence o
limT→∞ RRER
(
)
ϖT
(
)
d
; he con adic ion in §4.10 does no equi e such a limi .
Lemma A.55 p o ides
T
–uni o m bounds i s , and only hen passes o
α↓
0 o he
x– egula o . This is he only o de o egula o s used anywhe e in he pape .
F.4. Rela ion o he quad a ic– o m egula o (Appendix A). In he
sel –adjoin /closabili y analysis we some imes add a compac egula o in
x
,
RI,ε
:=
R
+
ε
1
I
(Lemma A.8), o ob ain local coe ci i y on
I
and hen send
ε↓
0by
mono one o ms. This egula o is independen o he EF molli ica ion abo e; he
o me ac s a he le el o he quad a ic– o m domain, he la e a he le el o EF
es ke nels. Bo h limi s commu e wi h all EF in e changes because o he blockwise
T–uni o m bounds and he con inui y in R.
Clay–compliance no e. All egula isa ions in his appendix ac on ex e nal es
ke nels
R
only;
ζ
and
ξ
a e ne e modi ied. The ze o se o
ξ
is he e o e un ouched
by any limi α↓0conside ed he e.
This comple es Appendix F.
88 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
Appendix G. Measu e– heo e ic lemmas
Aim. Cla i y he use o “almos e e ywhe e in
” language, he ea men o o dina es
o ze os, and he jus i ica ion o unca ion and window limi s a singula o dina es.
We also eco d measu abili y and in eg abili y p ope ies o he ene gy densi y ha
legi imise all Tonelli–Fubini and domina ed–con e gence s eps appea ing in he
p oo .
G.1. Coun abili y o ze o o dina es.
Lemma A.57 (Coun abili y o ze o o dina es).Le
Z:= {γ∈R:∃β∈(0,1) wi h ξ(β+iγ)=0}
be he se o o dina es o non i ial ze os o
ζ
(
s
). Then
Z
is coun able; hence i is
null o Lebesgue measu e d and o he Gaussian weigh s ϖT( )d (T > 0).
P oo .
Since
ξ
is en i e o ini e o de , i s ze os a e isola ed and ini e in any compac
se . Fo each
N∈N
, he se
{ρ
:
|ℜρ| ≤ N, |ℑρ| ≤ N}
is ini e; p ojec ing o
o dina es and aking he union o e
N
p oduces a coun able subse o
R
. Coun able
se s a e null o d , hence also o ϖTd (absolu e con inui y). □
G.2. Measu abili y, local in eg abili y in
, and p oduc measu e. Le
g(x, ) := log |ξ(x+i )|2and HR(x, ) := R(x)|∂xg(x, )|2 o R∈S0.
Lemma A.58 (Measu abili y and
L1
loc
in
).The map (
x,
)
7→ HR
(
x,
)is
Bo el–measu able on (0
,
1)
×R
. Fo a.e.
∈R Z
, he unc ion
x7→ HR
(
x,
)
belongs o L1
loc(0,1). Consequen ly, he R–weigh ed ene gy
ER( ) := Z1
0
HR(x, )dx ∈[0,∞]
is a measu able, locally in eg able unc ion o , i.e. ER∈L1
loc(R).
P oo .
On (0
,
1)
×
(
R Z
), he map (
x,
)
7→ ξ′
(
x
+
i
)
/ξ
(
x
+
i
)is analy ic, hence
con inuous; composing wi h
R
and absolu e– alue/squa ing p ese es measu abili y.
Fo any ixed
γ∈ Z
, le
{β
:
ξ
(
β
+
iγ
)=0
}
(a ini e se in (0
,
1)) be he abscissae
a o dina e
γ
. Ex end
HR
by se ing
HR
(
β, γ
) := +
∞
o each such
β
and keep he
analy ic de ini ion elsewhe e on he ho izon al line
=
γ
. This modi ica ion occu s
on a coun able union o e ical lines, hence p ese es Bo el measu abili y.
Fo
/∈ Z
, he e a e no
x∈
(0
,
1) wi h
ξ
(
x
+
i
) = 0; hus
x7→ ∂xg
(
x,
)is
smoo h. By Appendix B (Lemma A.18) we ha e he uni o m en elope
|∂xg
(
x,
)
| ≪ϵ
1+
log
(2+
| |
)on
x∈
[
ϵ,
1
−ϵ
]. Since
R∈S0
is smoo h, nonnega i e, and in eg able
on (0
,
1) wi h
R
(
x
)
≍
(
x−1
2
)
2
nea
x
=
1
2
, we ob ain
HR
(
·,
)
∈L1
loc
(0
,
1). (I
∈ Z
and some ze o occu s a
x
=
β
, he local singula i y
|∂xg
(
x,
)
|∼|x−β|−1
is
ha mless when
β
=
1
2
because
R
(
x
)
≍
(
x−1
2
)
2
cancels i ; o he wise we ega d he
poin
x
=
β
as assigned alue +
∞
and ea any needed in eg a ions by unca ion
in x.)
Fo
ER∈L1
loc
(
R
), ix
χ∈C∞
c
(
R
),
χ≥
0. By he en elope abo e (see also
Appendix B, Co . A.21),
ZR
ER( )χ( )d =Z(0,1)×R
HR(x, )χ( )dx d < ∞,
so ERis measu able and locally in eg able in .□
THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 89
Lemma A.59 (P oduc –measu e Tonelli–Fubini).Fix
T >
0and de ine he p oduc
measu e
dµR,T
(
x,
) :=
R
(
x
)
dx ⊗ϖT
(
)
d
. I
F≥
0is measu able on (0
,
1)
×R
,
hen
ZZ F dµR,T =Z1
0ZR
F(x, )ϖT( )d R(x)dx =ZRZ1
0
F(x, )R(x)dxϖT( )d .
I
|F| ≤ DT
(
)wi h
DT∈L1
(
R, ϖTd
), hen bo h i e a ed in eg als con e ge
absolu ely and coincide.
P oo .
Tonelli’s heo em applies in he nonnega i e case. Fo he signed case, use
domina ed con e gence wi h majo an
R
(
x
)
DT
(
)
ϖT
(
), which is in eg able on
(0,1) ×Rbecause R1
0R(x)dx < ∞and DT∈L1(R, ϖTd ).□
Use in he main ex : Lemma A.58 gua an ees ha
ER
(
)is a legi ima e (ex-
ended– eal) unc ion o
and ha he ime–windowed unc ionals a e bona ide
Lebesgue in eg als; Lemma A.59 is he p oduc –measu e o malism used h oughou
§4.11 and §4.10.
G.3. Lebesgue di e en ia ion in xand he lux densi y.
Lemma A.60 (Lebesgue di e en ia ion in x).Fo a.e. ∈Rand a.e. x0∈(0,1),
lim
ε↓0
1
2εZx0+ε
x0−ε
R(x)|∂xg(x, )|2dx =R(x0)|∂xg(x0, )|2.
In pa icula , a x0=1
2one has R(1
2) = 0 and hence he limi equals 0 o a.e. .
P oo .
By Lemma A.58, o a.e.
he unc ion
x7→ R
(
x
)
|∂xg
(
x,
)
|2
is locally
in eg able. Apply he Lebesgue di e en ia ion heo em in
x
. The alue a
x0
=
1
2
ollows since R(1
2) = 0.□
Use in he main ex : This is he igo ous o m o he poin wise lux densi y ex ac ion
used in §4.11 and, implici ly, in he de ini ion o he cylind ical lux nea
x0
=
β
( o away om Z).
G.4. T unca ion a ze o o dina es and Gaussian app oxima e iden i y in
.
Lemma A.61 (Mono one unca ion a
=
γ
).Le
γ∈ Z
and
F≥
0measu able on
R
. De ine
Iη
:=
R| −γ|>η F
(
)
ϖT
(
)
d
. Then
limη↓0Iη
exis s in [0
,∞
]. I
F≤DT
wi h
DT∈L1
(
R, ϖTd
), he limi is ini e and all in e changes wi h sums/in eg als
used in he pape a e jus i ied by domina ed con e gence.
P oo .
Mono one con e gence yields exis ence o he limi . The domina ion hypo h-
esis implies uni o m in eg abili y, which gi es he in e changes. □
Lemma A.62 (Gaussian app oxima e iden i y in
).Le
∈L1
loc
(
R
). Wi h he
mass–one Gaussians ϖT( )=(√πT)−1e− 2/T 2,
lim
T↓0ZR
( )ϖT( −τ)d = (τ) o a.e. τ∈R.
I is con inuous a τ, he con e gence holds wi hou he “a.e.” quali ie .
P oo .
The amily
{ϖT}T >0
is an app oxima e iden i y on
R
(e en, mass one, igh ).
Apply he Lebesgue di e en ia ion heo em in .□
96 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
1AIndica o o a se A⊂R. (Appendix A)
D. Ene gies, Lyapuno unc ional, measu es
ER( )
Weigh ed ho izon al ene gy
ER
(
) =
ZR
R
(
x
)
|∂xg
(
x,
)
|2dx ∈
[0,∞]. (§4.10)
ER,T
Windowed Lyapuno unc ional
ER,T
=
ZR
ER
(
)
ϖT
(
)
d
.
(§4.10)
dµR,T
P oduc measu e
dµR,T
(
x,
) :=
R
(
x
)
dx ⊗ϖT
(
)
d
. (Appen-
dix A)
Z
O dina es o non i ial ze os:
Z
=
{γ
:
∃β, ξ
(
β
+
iγ
) = 0
}
;
coun able se , Lebesgue-null in . (Appendix A)
E. Explici – o mula (EF) objec s and blocks
Φ(s)
Sho hand Φ(
s
) =
bφ

(
s−1
2
)
/i
used in EF manipula ions.
(Appendix A)
BΓ[R;T]
Gamma block in EF assembly ( e ical line in eg al wi h
Γ′/Γ). (§4.8, Appendix A)
BDE[R;T]
Di ichle –Eule (p ime) block in EF assembly. (§4.8, Appen-
dix A)
BZ[R;T]
Ze o–sum block (windowed sum o e non i ial ze os). (§4.8,
Appendix A)
ZR(ν, )
Ze o–block ke nel a e linea isa ion; see s uc u e
(A.20)
wi h
coe icien s κR. (Appendix A)
κR(ν;β)
F equency coe icien (Schwa z in
ν
) wi h cen al anishing
κR
(
ν
;
1
2
) = 0 and low– equency cancella ion
|κR
(
ν
;
β
)
| ≪R
|ν|. (Appendix A)
Kν(γ, γ′)
Time ke nel
Kν
(
γ, γ′
) =
1
π|ν|e−2π|ν||γ−γ′|
in he ze o–block
quad a ic o m. (Appendix A)
bγ
Localised ze o weigh (a ising om linea isa ion/windowing)
a ached o
ρ
=
β
+
iγ
; band
-
summed in
Bm
. (Appendix A)
Zm, Bm
Uni –band index
Zm
=
{γ
:
m≤γ < m
+ 1
}
and ene gy
p oxy B2
m=Pγ∈Zmb2
γ. (Appendix A)
CΓ
(
R
)
, CDE
(
R
)
, CZ
(
R
)
Blockwise cons an s in he EF bound, depending on ini ely
many
S
–semino ms o
R
, independen o
T
. (§4.8, Appen-
dices A,A)
C(R)
Assembled EF cons an
C
(
R
) =
CΓ
(
R
) +
CDE
(
R
) +
CZ
(
R
).
(§4.8)
F. Ze o coun ing and Li coe icien s
N(T)
Ze o coun ing unc ion
N
(
T
)=#
{ρ
=
1
2
+
iγ
: 0
< γ ≤T}
.
(§5.1, Appendix A)
N(u; 1)
Uni –band ze o coun on [
u, u
+ 1);
N
(
u
; 1)
≪log
(2 +
|u|
).
(Appendix A)
λn
Li’s coe icien s
λn
=
X
ρ
1
−
1
−1
ρn
(sum aken sym-
me ically). (Appendix A)
G. Gamma, p imes, and auxilia y a i hme ic

THE RIEMANN HYPOTHESIS AS A ZEROFLUX CONDITION 97
Γ(s)
Eule gamma unc ion;
ψ
(
z
)=Γ
′
(
z
)
/
Γ(
z
)(digamma). Ve i-
cal bounds in Appendix A. (§2.1)
Λ(n)
on Mangold unc ion: Λ(
n
) =
log p
i
n
=
pk
wi h
p
p ime,
k≥1, and Λ(n) = 0 o he wise. (§2.1)
aR(n)
P ime
-
block coe icien (linea in
b
R
) occu ing in he Di ich-
le –Eule block. (§4.8, Appendix A)
H. Measu e heo y and windows
dµR,T
P oduc measu e
R
(
x
)
dx ⊗ϖT
(
)
d
(duplica e o he en y
in D o con enience). (Appendix A)
χj
E en pa i ion o uni y in
wi h bounded o e lap (uni
-
band
localisa ion). (Appendix A)
δx
Di ac mass a
x
; e.g. a no malised amily
e
Rα⇒δ1/2
(weak
limi ). (Appendix A)
I. De i ed ke nels and ime in eg als
KR
E en, nonnega i e p incipal
-
pa ke nel nea a ze o; he asso-
cia ed weigh is WR(ρ;T)=(KR∗ϖT)(γ). (Appendix A)
Mε
Model cusp in eg al M
ε
(
a
) =
Zε
−ε
u2
(u2+a2)2du
wi h
a
=
| −γ|. (Appendix A)
J. Asymp o ic and inequali y no a ion
O(·), o(·)
Landau symbols; subsc ip s indica e pa ame e dependence,
e.g. OR(1). (Th oughou )
≪
Vinog ado no a ion:
A≪B
means
|A| ≤ C B
o an abso-
lu e o indica ed cons an C. (Th oughou )
≍
Two–sided bound up o cons an s depending only on ixed
pa ame e s. (Th oughou )
K. Auxilia y ope a o s (bookkeeping only)
Rx (R)
E en Schwa z-class weigh ob ained om explici – o mula
pai ing o
−ζ′/ζ
wi h admissible es unc ions; canonical
model
Rα
(
x
) = (
x−1
2
)
2e−α(x−1
2)2
. Used only as an ex e nal
p obe; ne e modi ies ζ. (§2.1–§2.2, App. C)
HC
“Hype –conjuga ion” ope a o (bookkeeping o weigh ed
slope dynamics); no used o ede ine ξ. (§2.3)
ERU
“Equa ion o Rela ional Uni y” umb ella symbol o measu e-
men amewo k ( egula o s/a e aging only). (§2.4)
No es. (i) All Fou ie /Planche el iden i ies use he con en ion o Appendix A.
(ii) Th oughou , he p oo s use he no malised window
ϖT
; i an unno malised
wT
appea s in an in e media e display, i is escaled by (
√πT
)
−1
in he inal es ima es.
(iii) EF cons an s
CΓ
(
R
)
, CDE
(
R
)
, CZ
(
R
)depend only on ini ely many
S
–semino ms
o R; see §4.8, Appendices A,A.
(i ) The admissible class
S0
equi es quad a ic anishing a
x
=
1
2
; no global
Fou ie
-
posi i i y assump ion is in oked anywhe e in he p oo (Appendix A, Re-
ma k).
98 PROF. ELIAHI PRIEST HON.DS.C(UFSEI)
Re e ences
[1]
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[2] A. I ić, The Riemann Ze a-Func ion: Theo y and Applica ions, Do e , 2003.
[3]
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[6]
G.H. Ha dy and J.E. Li lewood, “Con ibu ions o he heo y o he Riemann ze a- unc ion
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[7]
M. Reed and B. Simon, Me hods o Mode n Ma hema ical Physics II: Fou ie Analysis,
Sel -Adjoin ness, Academic P ess, 1975.
[8]
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1995.
[9]
A. Weil, “Su les ‘ o mules explici es’ de la héo ie des nomb es p emie s,” Comm. Sem. Ma h.
Uni . Lund (1952), 252–265.
[10]
A.E. Ingham, The Dis ibu ion o P ime Numbe s, Camb idge T ac s in Ma hema ics, no. 30,
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[11] T. Ka o, Pe u ba ion Theo y o Linea Ope a o s, Sp inge (1995).
[12]
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Sel -Adjoin ness, Academic P ess (1975).
[13]
M. Reed and B. Simon, Me hods o Mode n Ma hema ical Physics, ol. IV: Analysis o
Ope a o s, Academic P ess (1978).
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