PRH | Educ | 6.4 • A Gentle First Course in Adelic–Quantum Arithmetic

A Gen le Fi s Cou se in Adelic–Quan um A i hme ic
A p epa a o y, exe cise- ich in oduc ion
Aleksanda Pe išić
No embe 2025
Abs ac
This bookle o e s a gen le, linea -algeb a iendly pa h owa d he adelic–quan um (AQ)
iewpoin on a i hme ic. The slogan is ha addi ion beha es like a wa e (shi s), while
mul iplica ion beha es like a scale change (dila ions and p ime s eps). These wo
ac ions gene a e he adelic ax+bsymme y and sa is y he commu a ion ule
[Hidele, Hadele] = i Hadele ,
which is he adelic a a a o he classical [D, P] = iP on L2(R). We build up he pic u e om
i s p inciples—numbe s, p imes and composi es; ec o s and inne p oduc s; unc ions as
ec o s; Fou ie ans o m; a gen le look a p-adic numbe s; hen adeles/ideles and hei
basic ac ions—wi h many sho exe cises (odd-numbe ed answe s included).
How o ead his bookle
Each sec ion ends wi h bi e-size exe cises. I an idea eels ad anced, we pu a quick one-
pa ag aph Aside igh nex o i . You will need only: high-school algeb a, i s cou ses in
calculus and linea algeb a, and com o wi h complex numbe s. Measu e heo y is ligh ly used
and explained when needed.
Pa I
Wa m-up Ma hema ics
1 Numbe s, p imes, and composi es
De ini ion 1.1 (P ime and composi e).A numbe n∈N≥2is p ime i i s only posi i e di iso s
a e 1and n. I is composi e i n=ab wi h 1< a < n and 1< b < n.
Example 1.2. 2,3,5,7,11 a e p ime; 9 = 32and 49 = 72a e composi e.
Theo em 1.3 (Fundamen al Theo em o A i hme ic).E e y n≥2 ac o s uniquely as n=
Qpαi
iwi h pidis inc p imes, αi∈N.
Exe cise
W i e each numbe as a p oduc o p imes and lis i s p ime exponen s:2,9,5,49,360.
Then o m he ec o o exponen s (α2, α3, α5, α7, . . . ) o each. Which o hese ec o s
is spa se? Why?
1
Aside
[Supe posi ion idea] Think o a o mal basis {pk:pp ime, k ≥0}and a ach o an
in ege n=Qpαp he supe posi ion |ni=Ppαp|pii you only keep he p ime coun s.
La e we will e ine his o include all k≥1wi h weigh s.
Exe cise
(Pa i y and di isibili y.) Show nis e en i i s p ime-exponen ec o has α2≥1. How
would you ecognize di isibili y by 9in ha ec o ?
2 Vec o s, inne p oduc s, and complex ampli udes
A ec o space is a se whe e you can add ec o s and scale hem by numbe s. We use complex
numbe s because phases (angles) ma e in wa es.
De ini ion 2.1 (Inne p oduc ).On a complex ec o space V, an inne p oduc h·|·i is linea
in he i s slo , conjuga e-symme ic, and posi i e de ini e. I s induced no m is k k2=ph | i.
Example 2.2 (Column ec o s).Fo u, ∈Cn,hu| i=Pjuj j. The ma ix o a linea map
Tsa is ies hTu| i=hu|T∗ iwhe e T∗is he conjuga e anspose.
Exe cise
(Uni a y ma ices.) Show ha U∗U=Ii Up ese es inne p oduc s. Gi e an example
o a 2×2uni a y ha o a es ec o s by an angle.
3 Func ions as ec o s; he Fou ie ans o m
De ini ion 3.1. L2(R)is he space o squa e-in eg able unc ions wi h inne p oduc h |gi=
RR (x)g(x) dx.
P oposi ion 3.2 (Fou ie ans o m).The Fou ie ans o m b
(ξ) = RR (x)e−2πixξ dxis a
uni a y map L2(R)→L2(R).
Aside
[Why Fou ie ?] Shi s in xbecome phases in equency. This is why addi ion (shi s)
looks like wa es.
Exe cise
(Gaussian.) Show (x) = e−πx2is i s own Fou ie ans o m.
4 A 10-minu e iew o p-adics
De ini ion 4.1 (p-adic absolu e alue).Fo p ime p, w i e n=pkmwi h p-m. De ine
|n|p=p−kand ex end o Qby |a/b|p=|a|p/|b|p.
Aside
[In ui ion] |·|pmeasu es di isibili y by p. Numbe s wi h many p’s in hem look small
p-adically.
2
De ini ion 4.2. Qpis he comple ion o Qunde |·|p. I s in ege s Zp={x∈Qp:|x|p≤1}
o m a compac ing.
Exe cise
Compu e |9|3,|9|5, and |49|7. Which a e ≤1? Which a e <1?
5 Adeles and ideles (a iendly i s pass)
De ini ion 5.1 (Adeles).The adeles A=Q′
Q a e he es ic ed p oduc o he comple ions
o Q( eal =∞and all p-adics), whe e “ es ic ed” means almos all ini e componen s lie in
Zp.
De ini ion 5.2 (Ideles).The ideles A×a e he uni s o A, wi h mul iplica i e Haa measu e
d× and module | |A=Q | | .
Aside
[Takeaway] Adeles le us do addi ion and Fou ie analysis simul aneously o e he eal
line and all p-adic lines. Ideles do he same o mul iplica ion.
Exe cise
(Res ic ed p oduc .) Explain why he uple wi h x∞∈Rand xp∈Zp o all bu ini ely
many pde ines a poin in A.
Pa II
The ax+b s o y
6 The eal ax+b model and [D,P]=iP
Le U(a) (x) = (x−a)(shi ) and (V(λ) )(x) = λ−1/2 (x/λ)(dila ion). De ine gene a o s
P=−i ∂x, D =1
2(x∂x+∂xx) = −i
2(x P +P x).
Then on a na u al co e (Schwa z unc ions), [D, P ] = iP.
Exe cise
Ve i y [D, P] = iP by di ec compu a ion on e−πx2.
Aside
[Meaning] Scaling hen shi ing is no he same as shi ing hen scaling; he misma ch is
exac ly p opo ional o he shi gene a o P.
7 F om eal o adelic: he Hilbe space H
Le H=L2(A/Q,dx), he space o squa e-in eg able unc ions on adeles modulo a ionals. The
adelic affine g oup G=AoA×ac s by
(U(a, ) )(x) = | |−1/2
A  −1(x−a),(a, )∈A×A×.(1)
3
De ini ion 7.1 (Re e ence s a e).Take Φ = N Φ wi h Φ∞(x) = π−1/4e−πx2and Φp=1Zp.
Pe iodize o ΘΦ(x) = Pq∈QΦ(x+q)and no malize i in H.
Exe cise
(O hogonali y unde shi s.) Show hΘΦ|U(a, 1)ΘΦiis 1i a∈Qand decays apidly
o he wise.
8 Two lows: addi i e and idelic
Addi i e low. U( , 1) wi h gene a o
Hadele =id
d  =0
U( , 1).
A =∞ his is he usual momen um −i∂x∞; a ini e places i is he in ini esimal e sion o
x7→ x+ .
Idelic low. Spli A×in o he connec ed no m low and p ime s eps. Pick u∈A×wi h
| u|A=euand se
N=id
duu=0
U(0, u).
Fo each p ime p, le Tp=U(0, ϖp)mul iply by he idèle ha is pa place pand 1elsewhe e.
Fo mally w i e Tp=ei(log p)Np o de ine coun ing ope a o s Npon he p ime band. De ine
Hidele := N+X
p
(log p)Np.
Aside
[Domains] All commu a o s a e aken on he common in a ian co e o Schwa z–B uha
ec o s. S one’s heo em ensu es sel -adjoin gene a o s o uni a y lows.
Theo em 8.1 (Adelic ax+bcommu a o ).On he co e, [Hidele, Hadele] = i Hadele.
Exe cise
(Real place check.) Show ha a he eal place alone, [D, P] = iP ma ches he N–Hadele
pa o he ela ion.
Exe cise
(P ime s eps as phases.) Show Tpac s by a phase on Fou ie modes suppo ed on mul iples
o log pin he equency o A.
9 A hi d low: p ime–phase gauge (Helson wis )
Besides shi ing (addi i e low) and scaling/p ime s eps (idelic low), he e is a hi d, commu ing
uni a y low ha only ephases he p ime band. Fix eal phases {θp}pp ime and de ine
V( ) = expi Hgauge, Hgauge := X
p
θpNp, ∈R,
4
so ha on he oo h u=klog pone has
V( ) : δu−klog p7−→ ei k θpδu−klog p.
Equi alen ly, a he Eule –p oduc le el his is he Helson wis by a comple ely mul iplica i e
unimodula unc ion
χ (p) = ei θp, χ (n) = Y
pk∥n
χ (p)k, ζχ (s) = Y
p
1
1−χ (p)p−s=X
n≥1
χ (n)
ns.
P oposi ion 9.1 (Commu a ion wi h he wo lows).On he common Schwa z–B uha co e,
[Hgauge, Hadele] = 0,[Hgauge, Hidele] = 0.
Idea. Npac only on he disc e e p ime band, hence commu e wi h addi i e shi s. They
also commu e wi h he no m gene a o Nand wi h each o he , so Hgauge commu es wi h
Hidele =N+Pq(log q)Nq.
Aside
[Why phases?] Keeping |χ (p)|= 1 p ese es absolu e con e gence in Re s > 1and
comple e mul iplica i i y o Di ichle coefficien s. The gauge low ephases p ime powe s
bu does no ansla e hem (addi i e) o escale hem (mul iplica i e). Mo eo e , in he
Helson class one can p esc ibe ze o/pole se s in he hal -plane Re s < 1—in pa icula ,
ealize models whose non i ial ze os all lie on Re s=1
2—by a sui able choice o phases
a he p imes [7, 6, 5, 4].
Exe cise
(Semig oup law.) Show ha o he gauge low V( ) = exp(i Hgauge)one has V( +s) =
V( )V(s)and V(0) = I.
10 A ou h low: o us/Floque closu e
Pick a leng h L > 0and w ap he scale a iable u∈R o he ci cle
θu:= umod L∈S1
u∼
=R/LZ.
To “close he geome y” while keeping ull con ol, in oduce a Floque ( wis ) pa ame e α∈
R/2πZand he Floque pe iodiza ion ope a o
(ΠL,α )(θu) := X
n∈Z
einα (θu+nL).
This iden i ies u∼u+Lwi h quasi–pe iodic bounda y condi ion (u+L) = eiα (u). The
co esponding o us low is he 1–pa ame e uni a y g oup
W(α) := eiα w,so ha ΠL,α =W(α) ΠL,0,
whe e wis he (in ege ) winding ope a o eco ding how many w aps by Lwe shi in he
u–di ec ion.
P oposi ion 10.1 (Commu a ion wi h he h ee lows).On he Schwa z–B uha co e,
[W(α), Hadele ] = 0,[W(α), Hidele ] = 0,[W(α), Hgauge ] = 0.
5

Idea. Hadele and he connec ed pa o Hidele ac by ansla ions in u, which commu e wi h
pe iodiza ion ΠL,α. The disc e e p ime coun e s Np(hence Hgauge) ac only on he p ime band
and a e independen o u–w apping, so hey commu e wi h W(α)as well.
Aside
[Geome y in one line] Hadele slides/blu s along he base ci cle S1
u; Hidele eweigh s by
u(o pe o ms he hea ime change) be o e pe iodiza ion; Gauge spins he phase ci cle
a each p ime oo h. The o us low W(α)me ely ixes he “gluing ule” u∼u+L ia
he phase eiα.
Rema k 10.2 (Rela ion o ho izon al il ).On he o us, a eal ho izon al il edu co esponds
o a nonuni a y holonomy edL. To keep uni a i y in his ou h low, use α∈R(uni modulus).
I you need he eal il , wo k on he cylinde (u∈R) o ea i as a line-bundle wis wi h
monod omy edL.
11 P ime band and p ojec o s
Le Ω = {klog p:pp ime, k ∈N}be he “p ime band” in he mul iplica i e spec um. One
can de ine a ( o mal) p ojec o ΠΩ ha isola es hese equencies.
Exe cise
(De ec ing composi es.) A gue why ΠΩannihila es con ibu ions om p ime- ee in ege s
and explain how pkappea wi h weigh k.
12 A one- ec o pic u e ( he simples s o y)
Aside
S a e: a single ec o |ψi ∈ H =L2(A/Q).
Addi i e ac ion (wa e): |ψi 7→ U( , 1) |ψi; gene a o Hadele (like momen um).
Mul iplica i e ac ion (p ime/scale): |ψi 7→ U(0, )|ψi; gene a o Hidele (no m low
N+ p ime coun e s Np).
Key ela ion: doing a iny scale hen a iny shi di e s om shi hen scale by exac ly
i imes he shi : [Hidele, Hadele] = iHadele.
Example 12.1 (Toy supe posi ion).Conside he supe posi ion |χi=|2i+|9i+|5i+|49i.
The coun ing ope a o s sa is y N2|χi=|2i,N3|χi= 2 |9i,N5|χi=|5i,N7|χi= 2 |49i, o he s
0. Thus Hidele |χi= (log 2) |2i+ 2(log 3) |9i+ (log 5) |5i+ 2(log 7) |49i+N|χi.
Exe cise
(Weigh s.) Gene alize he example o |χi=Pn≤100 |niand compu e Pp(log p)Np|χi
explici ly in e ms o p ime powe s ≤100.
6
Pa III
A i hme ic eadou s (ideas, no p oo s)
13 Poisson summa ion and a quan um Weil iden i y
Pe iodizing he e e ence ec o ( o ming a he a unc ion) and using Poisson summa ion yields
an iden i y ma ching he classical Weil explici o mula, bu he e i appea s be o e any ac o -
iza ion in o p imes is pe o med.
Aside
[Ve y sho dic iona y] The a/Poisson ↔spec um ma ching; ze a ze os ↔ esonances;
a chimedean ac o ↔Gaussian en elope.
Re e ence s a e and addi i e ope a o s. Le H=L2(A/Q, dx)wi h he addi i e uni a y
U(a, 1) om (1). Take he ac o able e e ence ec o
Φ = O
Φ ,Φ∞(x) = π−1/4e−πx2,Φp=1Zp,
and pe iodize i o he he a s a e
ΘΦ(x) := X
q∈Q
Φ(x+q)∈ H,kΘΦk2= 1 (a e no maliza ion).
Fo a Schwa z–B uha es ∈ S(A)de ine he (addi i e) Weyl/Heisenbe g obse able
(W ψ)(x) := ZA
(y)ψ(x−y)dy (s ong in eg al on H),
and le Fbe he global addi i e Fou ie ans o m on L2(A/Q), uni a y wi h ke nel e−2πi⟨x,ξ⟩
buil om he s anda d local cha ac e s.
Aside
[Fou ie /Haa con en ions] The global addi i e Fou ie ans o m Fis aken wi h espec
o he s anda d sel -dual cha ac e
ψ(x) = ψ∞(x∞)Y
p
ψp(xp), ψ∞( ) = e−2πi , ψp(x) = e2πi {x}p,
whe e {x}p∈Q/Zis he p–adic ac ional pa . Haa measu es a e chosen sel -dual:
dx∞=Lebesgue on R(so
e−πx2=e−πξ2), dxpwi h ol(Zp) = 1 and b
1Zp=1Zp.
Wi h hese choices, he global measu e dx =Q dx makes Funi a y on L2(A/Q)and
he adelic Poisson summa ion holds wi hou ex a cons an s.
Theo em 13.1 (Quan um Weil iden i y).Fo he no malized he a s a e ΘΦand e e y ∈
S(A), he adelic expec a ions
µΦ( ) := hΘΦ|W ΘΦi, µ∨
Φ( ) := ΘΦF−1W FΘΦ
coincide:
µΦ( ) = µ∨
Φ( ).
This is a single adelic iden i y (a quan um Poisson/Weil s a emen ), holding be o e any placewise
ac o iza ion.
7
Aside
[Ope a o s in one line] W a e ages addi i e ansla ions agains ; conjuga ion by F
swaps ansla ion wi h modula ion. The heo em asse s he equali y o he wo expec a-
ion alues in he he a s a e ΘΦ.
Gaussian eadou (baseline)
To ex ac a i hme ic da a along he scale line, push o wa d a eal es on u= log | |A: ake
(u) = e−u2/(4 ), g (s) = ZR
(u)e(s−1
2)udu =√4π e (s−1
2)2.
The quan um iden i y yields he classical-looking eadou
X
ρ
g (ρ) = X
pX
k≥1
Λ(pk) (klog p) + A∞( ) + E( ), E( ) = 2√4π e /4.(2)
Readou A: Hadele + Hidele only
Two commu ing de o ma ions:
•Hadele (addi i e blu ). Con olu ion in uby Kσ(u) = 1
√2πσ e−u2/(2σ2)sends 7→ +σ2/2.
•Hidele (ho izon al blu ). The ho izon al hea eτ2
2∂2
ssends 7→ eff =
1−2τ2 ( o 0< <
1/(2τ2)).
Hence
X
ρ
e eff(ρ−1
2)2=X
p,k
Λ(pk)e− eff (klog p)2+A∞( eff) + 2√4π eff e eff /4.
Readou B: adding he gauge low ( hi d low)
Le χ (p) = ei θp(comple ely mul iplica i e, |χ (p)|= 1). Replace {ρ}by ze os {ρ }o F (s) =
Qp(1 −χ (p)p−s)−1. Then
X
ρ
e eff (ρ −1
2)2=X
p,k
Λ(pk)ei kθp
|{z}
gauge phase
e− eff (klog p)2+A∞( eff) + 2√4π eff e eff /4.
Readou C: all ou lows (wi h o us/Floque closu e)
Fix L > 0and a Floque pa ame e α∈R/2πZ. Pe iodize along he u–ci cle wi h quasi–pe iod
eiα:
K(L,α)
(θ) = X
n∈Z
exp−(θ+nL)2
4 einα, θ =θu−θu(k, p),
and include op ional s eng hs p∈(0,1] so pk= k
p. Then
X
ρ
e eff (ρ −1
2)2=X
p,k
Λ(pk) k
p
|{z}
s eng h
ei kθp
|{z}
gauge
K(L,α)
eff θu−(klog p) mod L
| {z }
o us closu e
+A∞( eff) + 2√4π eff e eff /4.
8
Exe cise
(Real Poisson.) P o e he one-dimensional Poisson summa ion o mula on R o Schwa z
unc ions and es i on he Gaussian.
14 Gaussian p obes and he RH signal
Phase-scanned Gaussian es ec o s can con e he p esence o any o -line ze o o ζ(s)in o
exponen ial g ow h on he p ime side. Ope a ionally, one seeks g ow h-de ec ion s a is ics
whose boundedness is equi alen o all non i ial ze os lying a Re(s) = 1
2.
Aside
[Cau ion] Tu ning hese ideas in o p oo s equi es ca e ul ope a o domains, spec al
decomposi ions, and ace-class manipula ions. He e we only ske ch he mechanisms.
Exe cise
(Saddle-poin eel.) On L2(R), show how a Gaussian window localizes equency and how
a small phase il shi s i s cen e . In e p e in e ms o de ec ing a nea by esonance.
A de i a i e-based RH p obe (wo ked example, m= 1)
Recall he Gaussian pai
(u) = e−u2/(4 ), g (s) = √4π e (s−1
2)2,
and he explici o mula eadou
X
ρ
g (ρ) = X
pX
k≥1
(log p)e− (klog p)2
| {z }
P( )
+A∞( ) + E( ), E( ) = 2√4π e /4.
De ine he Chebyshe hea momen s
H2m( ) := X
pX
k≥1
(log p) (klog p)2me− (klog p)2(m≥0),
so ha P(m)( ) = (−1)mH2m( ).
P oposi ion 14.1 (RH ⇒a i s de i a i e inequali y).Se Z( ) := Pρe (ρ−1
2)2. Unde RH,
(−1)mZ(m)( )≥0 o all m≥0, > 0.
In pa icula , o m= 1,
H2( )≥A′
∞( ) + E′( ) ( > 0).
Ske ch. Di e en ia e he eadou once:
Z′( ) = −H2( ) + A′
∞( ) + E′( ).
Unde RH, e e y ze o is s=1
2+iγ and Z′( ) = Pγγ2e− γ2≥0wi h a global minus sign, i.e.
Z′( )≤0. Hence H2( )≥A′
∞( ) + E′( ).
9
Exe cise
Ex.26 (In ini esimal dila ion). Le (Vλ )(x) = λ−1/2 (x/λ) o λ > 0. Show he
gene a o a λ=euis N=−i(x∂x+1
2).
Solu ion
Se u= log λ. Then Veu (x) = e−u/2 (e−ux). Di e en ia e a u= 0:d
du |0Veu =
−(1
2 +x ′). Thus N=id
du |0Veu=−i(x∂x+1
2).
Exe cise
Ex.27 (Commu a o wi h momen um). Wi h P=−i∂xand N om Ex.26, e i y
[N, P] = iP.
Solu ion
Use [x∂x, P] = [x, −i∂x]∂x+x[∂x, P] = i∂x. Also [1
2, P] = 0. Hence [N, P] = −i[i∂x] = iP .
Exe cise
Ex.28 (P ime coun e s in a oy basis). On he o mal basis {pk}, de ine Nppk=
kpkand Nqpk= 0 o q6=p. Compu e Hidele|2i+|9i+|5i+|49iigno ing he
smoo h N-pa .
Solu ion
(log 2) |2i+ 2(log 3) |9i+ (log 5) |5i+ 2(log 7) |49i.
Exe cise
Ex.29 (Poisson o a Gaussian amily). Show X
n∈Z
e−π n2= −1/2X
m∈Z
e−πm2/ o
> 0.
Solu ion
Apply Poisson summa ion o (x) = e−π x2, whose Fou ie ans o m is b
(ξ) =
−1/2e−πξ2/ . Then Pn (n) = Pmb
(m).
Exe cise
Ex.30 (Fou ie il /shi ). Show Fe2πiax (x)(ξ) = b
(ξ−a).
Solu ion
R (x)e2πiaxe−2πixξ dx =R (x)e−2πix(ξ−a)dx =b
(ξ−a).
Exe cise
Ex.31 (Uni a y and gene a o .) Show V( )is uni a y and ha id
d  =0V( ) = Hgauge
using he spec al ule Nppk=kpk.
16

Solu ion
On he p ime-powe basis {pk}we ha e Hgauge pk=θpkpkand hence V( )pk=
ei θpkpkwi h unimodula ac o ei θpk. The e o e kV( )pkk=kpkkand, by
linea i y/densi y, V( )is uni a y. Di e en ia ing V( ) = ei Hgauge a = 0 gi es
d
d  =0V( ) = i Hgauge, i.e. id
d  =0V( ) = Hgauge.
Exe cise
Ex.32 (Commu a o s anish.) Ve i y di ec ly on he p ime-powe basis ha
[Hgauge,Nq] = 0 o all q, and deduce [Hgauge, Hidele] = 0. Why does [Hgauge, Hadele] = 0
hold a e e y place?
Solu ion
Since he numbe ope a o s commu e, [Np,Nq] = 0 o all p, q, we ge [Hgauge,Nq] =
PpθpNp,Nq= 0. Thus
[Hgauge, Hidele] = [Hgauge, N +X
q
(log q)Nq]
= [Hgauge, N] + X
q
(log q)[Hgauge,Nq] = [Hgauge, N].
Bu Ngene a es he connec ed no m low and commu es wi h each p ime s ep Tp=
ei(log p)Np, so [N, Np] = 0 and hence [Hgauge, N] = 0. The e o e [Hgauge, Hidele] = 0.
Finally, Hadele ac s by addi i e ansla ions (di e en ia ion in he addi i e a iable), while
Hgauge ac s diagonally on he disc e e p ime band; hey ope a e on independen deg ees,
so [Hgauge, Hadele] = 0 on he Schwa z–B uha co e.
Exe cise
Ex.33 (F om phases o Eule ac o s.) S a ing om V( )ac ing by ei kθpon he
oo h klog p, de i e ha Qp(1 −χ (p)p−s)−1has Di ichle coefficien s χ (n)wi h χ
comple ely mul iplica i e and |χ (n)|= 1.
Solu ion
On each oo h klog p,V( )con ibu es a phase ei kθp, i.e. in Eule - ac o o m
1
1−χ (p)p−s=X
k≥0
χ (p)kp−ks, χ (p) := ei θp.
Mul iplying o e p imes and expanding, he coefficien o n=Qpkpis Qpχ (p)kp=
χ (n), which is comple ely mul iplica i e wi h |χ (n)|= 1. Thus Qp(1 −χ (p)p−s)−1=
Pn≥1χ (n)n−sas claimed.
Exe cise
Ex.34 (Tiny ini e example.) Fix θ2=π/3,θ3=−π/4and θp= 0 o p≥5. Compu e
he wis ac o s on 2,4,8,3,9,6,12 a ime = 1 ( emembe 6 = 2 ·3ge s χ(2)χ(3)).
17
Solu ion
Wi h θ2=π/3,θ3=−π/4, = 1:
χ(2) = eiπ/3=1
2+i√3
2, χ(3) = e−iπ/4=1
√2(1 −i).
Then
2 : χ(2) = eiπ/3,4 = 22:χ(2)2=ei2π/3,8 = 23:χ(2)3=eiπ =−1,
3 : χ(3) = e−iπ/4,9 = 32:χ(3)2=e−iπ/2=−i,
6 = 2 ·3 : χ(2)χ(3) = ei(π/3−π/4) =eiπ/12,
12 = 22·3 : χ(2)2χ(3) = ei(2π/3−π/4) =ei5π/12.
(Op ionally: eiπ/12 = cos(15◦) + isin(15◦)and ei5π/12 = cos(75◦) + isin(75◦).)
Exe cise
Ex.35 (Twis ed Gaussian iden i y.) Assume he un wis ed Gaussian explici o mula.
Inse he ac o ei kθpon he p ime side and explain why he same iden i y holds wi h
he ze o se eplaced by {ρ }(ze os o ζχ ).
Solu ion
S a om he un wis ed explici o mula wi h Gaussian p obe 0:
X
ρ
g 0(ρ) = X
p,k
Λ(pk) 0(klog p) + A∞( 0) + E( 0).
Replacing he Eule ac o s by (1 −χ (p)p−s)−1mul iplies he p ime-powe con ibu ions
by χ (p)k, so he p ime side becomes Pp,k Λ(pk)χ (p)k 0(klog p). By he same con-
ou /Poisson a gumen (Weil’s explici o mula applied o ζχ ), he le -hand side is he
sum o e ze os o ζχ , i.e. Pρ g 0(ρ ), while he a chimedean and endpoin e ms a e
unchanged ( he wis is a ini e p imes only). Hence he same iden i y holds wi h he
ze o se eplaced by {ρ }.
Exe cise
Ex.36 (Fi s mo ion o a ze o unde a gauge wis ). Le χ (p) = ei θpand F (s) =
ζχ (s) = Qp(1 −χ (p)p−s)−1. Assume ρis a simple ze o o ζ(so ζ(ρ) = 0 and ζ′(ρ)6= 0),
and le ρ be he ze o o F wi h ρ0=ρ. Show ha ρ′(0) = 0 and compu e
ρ′′(0) = −L1(ρ)2+L2(ρ), L1(s) = iX
p
θpp−s
1−p−s, L2(s) = −X
p
θ2
pp−s
(1 −p−s)2.
(Hin : expand F (s) om log F (s) = log ζ(s) + L1(s) + 2
2L2(s) + O( 3)and use he
implici unc ion heo em on F (ρ ) = 0.)
18
Solu ion
W i e H(s, ) := F (s)and en o ce H(ρ , )=0wi h ρ =ρ+δ( ),δ(0) = 0. F om
log F (s) = log ζ(s) + L1(s) + 2
2L2(s) + O( 3)we ge
F (s) = ζ(s)1 + L1(s) + 2
2L1(s)2+L2(s)+O( 3).
Taylo -expand in sa s=ρ:
ζ(ρ+δ) = ζ′(ρ)δ+1
2ζ′′(ρ)δ2+O(δ3), Lj(ρ+δ) = Lj(ρ) + O(δ).
Keep e ms up o 2and no e δ=ρ′(0) +1
2ρ′′(0) 2+O( 3). Then
0 = F (ρ+δ) = ζ′(ρ)δ+O(δ2)1 + L1(ρ) + 2
2L1(ρ)2+L2(ρ)+O( 3).
Collec coefficien s by powe s o : - A o de :ζ′(ρ)ρ′(0) = 0 ⇒ρ′(0) = 0 (since
ζ′(ρ)6= 0). - A o de 2: using ρ′(0) = 0, we ha e
0 = ζ′(ρ)1
2ρ′′(0) + 0 ·L1(ρ)+0 ⇒ρ′′(0) = −L1(ρ)2+L2(ρ).
(The las equali y comes om inse ing he 2expansion ac o 2
2L2
1+L2mul iplying
ζ′(ρ)δand ma ching coefficien s.) Thus ρ′(0) = 0 and ρ′′(0) = −L1(ρ)2+L2(ρ).
Exe cise
Ex. 37 (Floque pe iodiza ion and quasi–pe iodici y). Fix L > 0and α∈R.
De ine he Floque pe iodiza ion
(ΠL,α )(θ) := X
n∈Z
einα (θ+nL).
(a) Show ha g(θ) := (ΠL,α )(θ)sa is ies he quasi–pe iodic bounda y condi ion g(θ+
L) = eiαg(θ). (b) Con e sely, i gis de ined on [0, L)and obeys g(θ+L) = eiαg(θ)(by
ex ension), show g= ΠL,α wi h := g1[0,L).
Solu ion
Ex. 37. (a) Compu e
g(θ+L) = X
n∈Z
einα (θ+L+nL) = X
m∈Z
ei(m−1)α (θ+mL) = eiα g(θ).
(b) Fo θ∈[0, L)we ha e (ΠL,α )(θ) = (θ) = g(θ). Fo gene al θ, w i e θ=θ0+nL
wi h θ0∈[0, L); hen
(ΠL,α )(θ) = einα (θ0) = einαg(θ0) = g(θ),
using he quasi–pe iodici y o g. Hence g= ΠL,α .
19
Exe cise
Ex. 38 (Fou ie se ies o he pe iodized Gaussian ke nel). Fo > 0and L > 0,
de ine he Floque hea ke nel on he u–ci cle
K(L,α)
(θ) := X
n∈Z
exp−(θ+nL)2
4 einα.
Show ha K(L,α)
admi s he Fou ie se ies
K(L,α)
(θ) = 1
LX
m∈Z
exp− 2πm +α
L2expi2πm +α
Lθ.
(Hin : Apply Poisson summa ion o x7→ e−x2/(4 )eiαx/L e alua ed on he la ice x=
θ+nL.)
Solu ion
Ex. 38. Conside F(x) = e−x2/(4 )eiαx/L on R. Poisson summa ion gi es
X
n∈Z
F(θ+nL) = 1
LX
m∈Zb
F2πm
Lei2πm
Lθ.
A di ec Gaussian ans o m yields b
F(ξ) = √4π exp− (ξ−α/L)2. Plugging ξ=2πm
L
gi es X
n
e−(θ+nL)2/(4 )einα =1
LX
m∈Z
exp− 2πm +α
L2ei2πm+α
Lθ,
which is he claimed se ies.
Exe cise
Ex. 39 (Semig oup and hea equa ion on he u–ci cle). Le ∗deno e con olu ion
on he ci cle o leng h L:( ∗g)(θ) = 1
LRL
0 (θ−η)g(η)dη. Show ha
K(L,α)
1∗K(L,α)
2=K(L,α)
1+ 2( 1, 2>0),
and ha o any quasi–pe iodic g0wi h g0(θ+L) = eiαg0(θ), he unc ion g(θ, ) =
(K(L,α)
∗g0)(θ)sol es ∂ g=∂2
θgon he ci cle wi h he same quasi–pe iodic bounda y
condi ion.
20
Solu ion
Ex. 39. Using he Fou ie se ies om Ex. 38,
K(L,α)
(θ) = 1
LX
m∈Z
e− λ2
meiλmθ, λm:= 2πm +α
L.
Then con olu ion mul iplies Fou ie coefficien s, so
K 1∗K 2(m) = d
K 1(m)d
K 2(m) = e− 1λ2
me− 2λ2
m=e−( 1+ 2)λ2
m,
which equals
K 1+ 2(m). Fo he hea equa ion,
∂ g=X
m
(−λ2
m)e− λ2
mbg0(m)eiλmθ=∂2
θX
m
e− λ2
mbg0(m)eiλmθ=∂2
θg.
Each mode eiλmθobeys eiλm(θ+L)=eiαeiλmθ, hence g(θ, )is quasi–pe iodic wi h pa am-
e e α.
Exe cise
Ex. 40 (Fou h low commu es wi h he o he h ee). Le W(α)be he o us/Flo-
que closu e ope a o implemen ing u∼u+Lwi h phase eiα (equi alen ly, ac ing ia
ΠL,α on es unc ions). Show on he common co e ha
[W(α), Hadele] = [W(α), Hidele] = [W(α), Hgauge] = 0.
(Hin : Hadele gene a es ansla ions in u;Hidele spli s in o a connec ed no m low in uand
p ime coun e s independen o u;Hgauge ac s only on he p ime band.)
Solu ion
Ex. 40. T ansla ions in ucommu e wi h pe iodiza ion and wi h he quasi–pe iodic bound-
a y condi ion, hence [W(α), Hadele] = 0. The connec ed pa o Hidele is a (loga i hmic)
dila ion ac ing as a ansla ion in ua e aking logs, so i also commu es wi h W(α); he
disc e e p ime coun e s Npdo no ouch he u– a iable, hence [W(α),Pp(log p)Np] = 0,
gi ing [W(α), Hidele] = 0. Finally, Hgauge =PpθpNpac s only on he p ime band,
independen o u, so [W(α), Hgauge] = 0.
Exe cise
Ex. 41 (Aliasing on he o us: a conc e e compu a ion). Take L=
log 8. Compu e he posi ions on he u–ci cle θu(k, p) := (klog p) mod L o (p, k)∈
{(2,1),(2,2),(2,3),(3,1)}. Wi h phases θ2=π/3,θ3=−π/4, = 1, and s eng hs
2= 3= 1, lis he ou o us poin s (θu, ϕ)whe e ϕ= k θpmod 2π. Which ee h alias
a he same θu?
21

Solu ion
Ex. 41. Since L= log 8 = 3 log 2:
θu(2,1) = log 2, θu(2,2) = 2 log 2, θu(2,3) = 3 log 2 ≡0 (modL).
Fo p= 3,θu(1,3) = log 3 which is no a mul iple o log 2, so i is dis inc mod L. Phases
a = 1:
ϕ(2,1) = π
3, ϕ(2,2) = 2π
3, ϕ(2,3) = π, ϕ(3,1) = −π
4(mod 2π).
Thus he ou o us poin s a e
(log 2, π/3),(2 log 2,2π/3),(0, π),(log 3 mod log 8,−π/4).
Aliasing occu s because 3 log 2 ≡0 ( mod L), so he oo h (p, k) = (2,3) lands a he same
θuas (p, k)=(any,0) (i.e. he o igin o he u–ci cle). The o he h ee posi ions a e
dis inc .
Fu he eading and e e ences
Re e ences
[1] J. T. Ta e, Fou ie Analysis in Numbe Fields and Hecke’s Ze a-Func ions, Ph.D. hesis,
P ince on, 1950; ep in ed in J. W. S. Cassels and A. F öhlich (eds.), Algeb aic Numbe
Theo y, Academic P ess, 1967, pp. 305–347.
[2] A. Weil, Basic Numbe Theo y, 2nd ed., Sp inge , 1974.
[3] G. B. Folland, A Cou se in Abs ac Ha monic Analysis, 2nd ed., CRC P ess, 2016. (Fo
physics-s yle analogies: he ax+bg oup and i s uni a y ep esen a ions.)
[4] H. Helson, Compac g oups and Di ichle se ies,A k. Ma . 8(1969), 139–143. (Classical
s a ing poin o Di ichle se ies wi h unimodula comple ely mul iplica i e coefficien s.)
[5] K. Seip, Uni e sali y and dis ibu ion o ze os and poles o some ze a unc ions,J. Anal.
Ma h. 141 (2020), 331–381; a Xi :1812.11729. (Founda ional analysis o Helson ze a ze-
os/poles and uni e sali y.)
[6] I. Bochko and R. Romano , On ze oes and poles o Helson ze a unc ions,
a Xi :2106.15949, 2021. (A bi a y ze os/poles in 21/40 <Re s < 1uncondi ionally; in
1/2<Re s < 1unde RH.)
[7] J. Ande sson, Mi ag–Le le ype heo ems o Helson ze a– unc ions, a Xi :2408.15713,
2024. (Me omo phic con inua ion wi h p esc ibed ze os/poles in Re s < 1 o Helson ze as.)
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