A Short Telescoping Proof of Hata's Formula for the Euler-Mascheroni Constant

#A90 INTEGERS 25 (2025)
A SHORT TELESCOPING PROOF OF HATA’S FORMULA FOR
THE EULER-MASCHERONI CONSTANT
Niki a Kalinin
Ma hema ics and Compu e Science Depa men , Guangdong Technion Is ael
Ins i u e o Technology, Shan ou, China
[email p o ec ed]
Recei ed:7/17/25 , Re ised: 8/21/25, Accep ed: 9/17/25, Published: 11/5/25
Abs ac
This no e p esen s an elemen a y p oo o a o mula o Eule ’s cons an γ, o iginally
due o Masayoshi Ha a. We ob ain Ha a’s o mula by a elescoping a gumen o e
Fa ey in e als.
1. In oduc ion
Recall ha Eule ’s cons an (also known as he Eule –Masche oni cons an ) is de-
ined as
γ= lim
n→∞ 1 + 1
2+· · · +1
n−ln n.
Al e na i e se ies ep esen a ions include
γ=
∞
X
k=2
(−1)kζ(k)
k= log 4
π+
∞
X
m=2
(−1)m
m2m
m
X
k=1 m
k.
Eule ’s cons an is also ela ed o he Riemann ζ- unc ion h ough he iden i y
γ= log 4π+X
ρ
1
ρ−2,
whe e he sum is o e he non- i ial ze os ρo he Riemann ze a unc ion. Fu he
o mulas and connec ions can be ound in he su ey [1] and in a mo e elabo a e
su ey [4].
Howe e , hese and many o he p esen a ions o γo e li le hope o p o ing
ha γis an i a ional numbe (which is widely belie ed), so new o mulas and
me hods a e highly desi able. In his no e, we in es iga e a lesse -known se ies o
γ, due o Masayoshi Ha a.
DOI: 10.5281/zenodo.17535156
INTEGERS: 25 (2025) 2
De ini ion 1. An in e al a
b,c
dsuch ha
a, b, c, d ∈Z≥0,0≤a
b<c
d≤1,and ad −bc =−1
is called a Fa ey in e al. In he li e a u e, Fa ey in e als a e also o en called
S e n-B oco in e als. Le Fdeno e he se o Fa ey in e als. Le
F∗=0
1,1
n:n∈N⊂ F.
Masayoshi Ha a p o ed he ollowing heo em.
Theo em 1 ([2]).In he abo e no a ion
γ=1
2+1
2X
[a
b,c
d]∈F F∗
1
abcd(a+c)(b+d).
Ha a’s p oo is e y nice. He s udies p esen a ions o unc ions in a ce ain
Schaude basis associa ed wi h Fa ey in e als. Then, using a Pa se al- ype iden-
i y o he unc ion ψ( ) = {1
}1− {1
}, he de i es he abo e heo em. In
subsequen wo k, Haynes–Vaale [3] showed ha he de i a i es o hese piecewise-
linea unc ions om ha Schaude basis o m an o hono mal basis and a comple e
sys em o ma ingale di e ences in L2([0,1]), wi h applica ions o me ic p ope ies
o con inued ac ions.
The goal o his no e is o gi e a sho and elemen a y p oo o he abo e heo em
using he elescoping s uc u e o e Fa ey in e als.
2. A Telescoping P oo o Ha a’s Theo em
We begin wi h he ollowing lemma, which is p o ed by di ec compu a ion.
Lemma 1. Le I=a
b,c
dbe a Fa ey in e al. Then
ad +bc
abcd −a(b+d) + b(a+c)
ab(a+c)(b+d)−(a+c)d+ (b+d)c
(a+c)(b+d)cd
=(bc −ad)2
abcd(a+c)(b+d)=1
abcd(a+c)(b+d).
The iden i y in Lemma 1 sugges s o associa e he e m ad+bc
abcd o he in e al I,
so deno e
ha
b,c
di= (I) = ad +bc
abcd =1
bc +1
ad.
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The o he wo e ms in Lemma 1 co espond o he alues o (Ile ) and (I igh )
o wo Fa ey in e als Ile and I igh om he subdi ision o Iby he median a+c
b+d
o wo ac ions a
band c
d:
Ile =a
b,a+c
b+dand I igh =a+c
b+d,c
d.
Hence, Lemma 1 can be e o mula ed as
1
abcd(a+c)(b+d)= (I)− (Ile )− (I igh ).(1)
The idea o he p oo o Theo em 1 is o sum he Equa ion (1) o e all I∈ F F∗.
To illus a e he elescoping p ope y, we conside he ollowing example, whe e we
sum Equa ion (1) o e I=1
2,1
1wi h Ile =1
2,2
3and I igh =2
3,1
1:
1
2,1
1− 1
2,2
3− 2
3,1
1
+ 1
2,2
3− 1
2,3
5− 3
5,2
3
+ 2
3,1
1− 2
3,3
4− 3
4,1
1
= 1
2,1
1− 1
2,3
5− 3
5,2
3− 2
3,3
4− 3
4,1
1,
which is he seed e m 1
2,1
1minus he sum o bounda y e ms co esponding
o he pa i ion 1
2,3
5,2
3,3
4,1
1o 1
2,1
1.
To compu e he limi o ini e sums as abo e, in he ollowing lemma we show ha
he sums o bounda y e ms con e ge o he in eg al o 2
as he mesh o pa i ion
ends o ze o.
Lemma 2. Le I=a
b,c
dbe a Fa ey in e al wi h a pa i ion
a
b=a1
b1
=x1<a2
b2
=x2<· · · <am
bm
=c
d=xm,
such ha [ak
bk,ak+1
bk+1 ]is a Fa ey in e al o each k= 1, . . . , m −1and a > 0. Then

m−1
X
k=1
ak
bk
,ak+1
bk+1 −
c/d
Z
a/b
2
d 
≤
c
d−a
b·b2
a2·max
k=1,...,m−1
1
bkbk+1
.(2)
P oo . The bound on he igh o Inequali y (2) will ollow om he s anda d es i-
ma e 
(y−x)g(x) + g(y)
2−
y
Z
x
g( )d 
≤|y−x|2
2max
∈[x,y]|g′( )|,(3)
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o a C1- unc ion gon an in e al [x, y]. We conside g( ) = 1/ . Each Fa ey
in e al [xk, xk+1] = hak
bk,ak+1
bk+1 ihas leng h xk+1 −xk=1
bkbk+1 . The unc ion can
be ew i en as
ak
bk
,ak+1
bk+1 =1
bkbk+1 bk+1
ak+1
+bk
ak= (xk+1 −xk)·(g(xk) + g(xk+1)).
The e o e
m−1
X
k=1
ak
bk
,ak+1
bk+1 =
m−1
X
k=1
(xk+1 −xk)·(g(xk) + g(xk+1)),(4)
which con e ges o 2
c/d
R
a/b
g( )d as maxk|xk+1 −xk| → 0, since i is a Riemann sum.
Applying Inequali y (3) wi h g( )=1/ on each subin e al [xk, xk+1] and summing
o e k, hen using Equa ion (4), we ob ain

m−1
X
k=1
ak
bk
,ak+1
bk+1 −
c/d
Z
a/b
2
d 
≤
m−1
X
k=1 
ak+1
bk+1
−ak
bk
2
·max
∈[a
b,c
d]|g′( )|.
Finally, o es ablish Inequali y (2), we compu e max ∈[a
b,c
d]|g′( )|=b2
a2and
m−1
X
k=1 
ak+1
bk+1
−ak
bk
2
≤max
k=1,...,m−1
ak+1
bk+1
−ak
bk
·
c
d−a
b= max
k=1,...,m−1
1
bkbk+1
·
c
d−a
b.
We ecall some classical p ope ies o Fa ey in e als. Le
F0=0
1,1
1,F1=0
1,1
2,1
2,1
1,
F2=0
1,1
3,1
3,1
2,1
2,2
3,2
3,1
1, . . .
whe e Fn+1 is ob ained om Fnby eplacing each in e al I= [a
b,c
d]∈ Fnby
wo in e als [a
b,a+c
b+d] and [a+c
b+d,c
d]. Using induc ion by n, we see ha o each
[a
b,c
d]∈ Fnwe ha e
max(b, d)≥n+ 1.(5)
Since [0
1,1
1] is a Fa ey in e al and a(b+d)−b(a+c) = ad −bc, i ollows ha Fn
consis s o Fa ey in e als o each n≥0. Since he leng h o I= [a
b,c
d]∈ Fnis 1
bd ,
using Inequali y (5) we ge he ollowing co olla y.
Co olla y 1. The leng h o each in e al I∈ Fnis a mos 1
n+1 .
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We ecall he p oo ha S∞
n=0 Fn=F. I ollows om he cons uc ion ha
all Fna e disjoin . I emains o show ha each Fa ey in e al [a
b,c
d] belongs
o a ce ain Fn. We p oceed by induc ion on max(b, d). I max(b, d) = 1 hen
b=d= 1, a = 0, and c= 1, so [a
b,c
d] = [0
1,1
1]∈ F0.
Assume ha he claim holds o max(b, d)< k. Conside a Fa ey in e al [a
b,c
d]
wi h max(b, d) = k. Since ad −bc =−1, i is no possible ha b=d=k > 1.
Wi hou loss o gene ali y, suppose ha b < d. Then a<c. Thus c
dis he median
o ac ions a
band c−a
b−d; also a(b−d)−b(c−a) = ad −bc =−1; so ha
b,c−a
b−diis
a Fa ey in e al. Since max(b, b −d)< k, by induc ion hypo hesis we ha e ha
[a
b,c−a
b−d]∈ Fn−1 o a ce ain n≥1. Hence [a
b,c
d]∈ Fnby he de ini ion.
Co olla y 2. Fo each in e al I= [a
b,c
d]∈ F F∗ he e exis s n≥1such ha
I⊂[1
n+1 ,1
n].
No e ha Icanno be [0
1,1
1], so Imus be con ained in [0
1,1
2] o in [1
2,1
1] =
[1
n+1 ,1
n] o n= 1. I Iis con ained in [0
1,1
2], hen, since Icanno be equal o [0
1,1
2],
Imus be con ained in [0
1,1
3] o [1
3,1
2]=[ 1
n+1 ,1
n] o n= 2, e c.
Lemma 3. Fix n≥1. Conside he in e al Jn=h1
n+1 ,1
ni∈ Fn. Then he sum
o exp essions in Equa ion (1) o e all Fa ey in e als I⊂Jnequals
1
n+1
n+ 1 −2 ln(n+ 1
n).
P oo . Sum Equa ion (1) o e he in e als I=h1
n+1 ,1
ni,h1
n+1 ,2
2n+1 i,h2
2n+1 ,1
ni,
e c., each ime subdi iding all in e als by he median o i s endpoin s. A e N
s eps we ha e a pa i ion o Jnas in Lemma 2. The e o e
X
I∈SN
i=0 Fn+i,
I⊂Jn (I)− (Ile )− (I igh )=  1
n+ 1,1
n−
2N
X
k=1
ak
bk
,ak+1
bk+1 ,
(6)
whe e {hak
bk,ak+1
bk+1 i}2N
k=0 ={I⊂Fn, I ∈ Fn+k}. Due o Inequali y (2) and Co ol-
la y 1, as N→ ∞, Equa ion (6) con e ges o
 1
n+ 1,1
n−
1
n
Z1
n+1
2
d =1
n+1
n+ 1 −2 ln(n+ 1
n).
P oo o Theo em 1. Since each in e al in F F∗is con ained in one o he in e als
Jn= [ 1
n+1 ,1
n], using he elescoping iden i y om Lemma 1 and Lemma 3 o each

INTEGERS: 25 (2025) 6
o he in e als Jn, we can ea ange he summands so ha o each k, we g oup all
e ms co esponding o he in e als con ained in [ 1
k+1 ,1], compu e hei sum, and
hen le k→ ∞. The e o e, we spli he sum om Theo em 1 as
1
2+1
2lim
k→∞ k
X
n=1 1
n+1
n+ 1−2 ln n+ 1
n!,
which e alua es o
1
2·2 lim
k→∞ k+1
X
n=1
1
n−ln(k+ 1) −1
2(k+ 1)!=γ.
Acknowledgemen . I would like o hank he e iewe whose sugges ions im-
p o ed his a icle.
Re e ences
[1] T. P. Dence and J. B. Dence, A su ey o Eule ’s cons an , Ma h. Mag. 82 (4) (2009), 255–265.
[2] M. Ha a, Fa ey ac ions and sums o e cop ime pai s, Ac a A i h. 70 (2) (1995), 149–159.
[3] A. K. Haynes and J. D. Vaale , Ma ingale di e ences and he me ic heo y o con inued
ac ions, Illinois J. Ma h. 52 (1) (2008), 213–242.
[4] J. Laga ias, Eule ’s cons an : Eule ’s wo k and mode n de elopmen s, Bull. Ame . Ma h.
Soc. (N.S.) 50 (4) 2013, 527–628.