Volume 2009, Article ID 626489,22pages doi:10.1155/2009/626489
Research Article
On Rational Approximations to Euler’s Constant
γ
and to
γ
log
a/b
Carsten Elsner
Fachhochschule f ¨ur die Wirtschaft Hannover, Freundallee 15, 30173 Hannover, Germany
Correspondence should be addressed to Carsten Elsner,[email protected]
Received 4 December 2008; Accepted 13 April 2009
Recommended by St´ephane Louboutin
The author continues to study series transformations for the Euler-Mascheroni constantγ. Here, we discuss in detail recently published results of A. I. Aptekarev and T. Rivoal who found rational approximations toγandγlogqq∈Q>0defined by linear recurrence formulae. The
main purpose of this paper is to adapt the concept of linear series transformations with integral coefficients such that rationals are given by explicit formulae which approximateγandγlogq. It is shown that for everyq∈ Q>0and every integerd ≥ 42 there are infinitely many rationals
am/bmform1,2, . . .such that|γlogq−am/bm| 1−1/dd/d−14d m
andbm|Zmwith logZm∼12d2m2formtending to infinity.
Copyrightq2009 Carsten Elsner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let
sn:
1
1
1
2
1 3· · ·
1 n−1
−logn n≥2. 1.1
It is well known that the sequencesnn≥1converges to Euler’s constantγ0,577. . ., where
snγO
1 n
n≥1. 1.2
Nothing is known on the algebraic background of such mathematical constants like Euler’s
constant γ. So we are interested in better diophantine approximations of these numbers,
particularly in rational approximations.
In 1995 the author1introduced a linear transformation for the seriessnn≥1 with
integer coefficients which improves the rate of convergence. Letτ be an additional positive
Proposition 1.1see1. For any integersn≥1 andτ ≥2 one has
n
k0 −1nk
nkτ−1 n
n
k
·skτ−γ
≤
τ−1!
2nn1n2· · ·nτ. 1.3
Particularly, by choosingτn≥2, one gets the following result.
Corollary 1.2. For any integern≥2one has
n
k0 −1nk
2nk−1 n
n
k
·snk−γ ≤
1
2n22n
n
≤ 1
n3/2·4n. 1.4
Some authors have generalized the result ofProposition 1.1under various aspects. At
first one cites a result due to Rivoal2.
Proposition 1.3see2. Forntending to infinity, one has
γ−
1
−2n
n
k0 −1k
2n2k n
n
k
s2kn1
O
1 n27n/2
. 1.5
Kh. Hessami Pilehrood and T. Hessami Pilehrood have found some approximation
formulas for the logarithms of some infinite products including Euler’s constant γ. These
results are obtained by using Euler-type integrals, hypergeometric series, and the Laplace
method3.
Proposition 1.43. Forntending to infinity the following asymptotic formula holds:
γ−
n
k0 −1nk
nk n
n
k
skn1
4n1on. 1.6
Recently the author has found series transformations involving three parametersn,τ1
andτ2,4. In Propositions1.5and1.6certain integral representations of thediscreteseries
transformations are given, which exhibit importantanalytical tools to estimate the error
terms of the transformations.
Proposition 1.5see4. Letn≥1,τ1≥1,andτ2 ≥1 be integers. Additionally one assumes that
Then one has
n
k0 −1nk
nτ1k
n
n
k
·skτ2−γ
−1n1
1
0
1
1−u
1 log u
·uτ2−τ1−1· ∂
n
∂un
unτ11−un n!
du.
1.8
Proposition 1.6see4. Letn≥1,τ1≥1 andτ2≥1 be integers. Additionally one assumes that
1τ1 ≤ τ2 ≤ 1nτ1. 1.9
Then one has
n
k0 −1nk
nτ1k
n
n
k
·skτ2−γ
−1nτ2−τ1
1
0
1
0
wt·1−u
nτ1un1−tτ2−τ1−1tnτ1−τ21
1−utn1 du dt,
1.10
with
wt: 1
t·
π2log2
1 t −1
. 1.11
Setting
nτ2dm, τ1 d−1m−1, d≥2, 1.12
one gets an explicit upper bound fromProposition 1.6
Corollary 1.7. For integersm≥2,d≥3, one has
dm
k0
−1dmk
2d−1mk−1 dm
dm
k
·skdm−γ < Cd·
1−1/dd
d−14d m−2
, 1.13
where 0< Cd≤1/16π2is some constant depending only ond. Ford2 one gets
2m
k0 −1k
3mk−1
2m
2m
k
·sk2m−γ <
16 7π
2
· 1
For an application ofCorollary 1.7let the integersBmandAmbe defined by
Bm:l.c.m.1,2,3, . . . ,4m,
Am:Bm
2m
k0 −1k
3mk−1
2m
2m
k
·
1 1
2· · · 1 k2m−1
.
1.15
Λkdenotes the von Mangoldt function. By [5, Theorem 434] one has
ψm:
k≤m
Λk∼m. 1.16
Then, forε: log 55/4−1>0.0018, there is some integerm0such that
Bmeψ4m < e41εm55m m≥m0. 1.17
Multiplying1.14byBm, we deduce the following corollary.
Corollary 1.8. There is an integerm0such that one has for all integersm≥m0that
Bm
2m
k0 −1k
3mk−1
2m
2m
k
·logk2m γBm−Am <
16 7π
2
·
55 64
m
. 1.18
2. Results on Rational Approximations to
γ
In 2007, Aptekarev and his collaborators6found rational approximations toγ, which are
based on a linear third-order recurrence. For the sake of brevity, letDn l.c.m.1,2, . . . , n.
Proposition 2.1see6. Letpnn≥0andqnn≥0be two solutions of the linear recurrence
16n−15n1un1
128n340n2−82n−45 un
−n256n3−240n264n−7 un−1nn−116n1un−2
2.1
withp00,p12,p231/2 andq01,q13,q225. Then, one hasqn∈Z,Dnpn∈Z, and
γ−pn
qn ∼c0e−2
√
2n, q n∼ c1
n1/4 2n!
n! e
√
2n, 2.2
It seems interesting to replace the fractionpn/qnby
An
Bn :
Dnpn
Dnqn, 2.3
and to estimate the remainder in terms ofBn.
Corollary 2.2. Let 0< ε <1. Then there are two positive constantsc2, c3, such that for all sufficiently large integersnone has
c2exp
−21ε√2
logBn/log logBn
<γ−An Bn
< c3exp
−21−ε√2
logBn/log logBn
.
2.4
Recently, Rivoal 7 presented a related approach to the theory of rational
approxi-mations to Euler’s constantγ, and, more generally, to rational approximations for values of
derivatives of the Gamma function. He studied simultaneous Pad´e approximants to Euler’s functions, from which he constructed a third-order recurrence formula that can be applied to
construct a sequence inQzthat converges subexponentially to logz γfor any complex
numberz∈C\−∞,0. Here, log is defined by its principal branch. We cite a corollary from
7.
Proposition 2.3see7. (i) The recurrence
n328n118n19Un3
24n2145n215 8n11U
n2
−24n3105n2124n25 8n27Un1
n228n198n27Un,
2.5
provides two sequences of rational numberspnn≥0andqnn≥0withp0−1,p14,p277/4 and
q01,q1 7,q265/2 such thatpn/qnn≥0converges toγ. (ii) The recurrence
n1n2n3Un3
3n219n29 n1Un2
−3n36n2−7n−13 U
n1 n23Un,
2.6
provides two sequences of rational numberspnn≥0andqnn≥0withp0 −1,p1 11,p271 and
The goal of this paper is to construct rational approximations toγloga/bwithout using recurrences by a new application of series transformations. The transformed sequences of rationals are constructed as simple as possible, only with few concessions to the rate of
convergencesee Theorems2.4and6.2below.
In the following we denote byB2nthe Bernoulli numbers, that is,B21/6,B4−1/30,
B6 1/42, and so onIn Sections3–6the Bernoulli numbers cannot be confused with the
integersBnfromCorollary 2.2.In this paper we will prove the following result.
Theorem 2.4. Leta≥1,b≥1,d≥42 andm≥1 be positive integers, and
Sn: an−1
j1
1 j −
bn−1
j1
1 j 2
n−1
j1
1 j −
n2−1
j1
1
j −
1 2n2
dm
j1
B2j
2j
1 n2j
1 a2j −
1 b2j 1
− 1
n4j
, n≥1.
2.7
Then,
dm
k0
−1dmk
2d−1mk−1 dm
dm
k
Skdm−γ−loga
b
< c4·
1−1/dd
d−14d m
, 2.8
wherec4is some positive constant depending only ond.
3. Proof of
Theorem 2.4
Lemma 3.1. One has for positive integersdandm
gk:
2d−1mk−1 dm
dm
k
<16dm 0≤k≤dm. 3.1
Proof. Applying the well known inequalityg
h ≤2
g, we get
2d−1mk−1 dm
dm
k
≤22d−1mdm−12dm24dm−m−1< 16dm. 3.2
This proves the lemma.
gktakes its maximum value forkk0with
k0
√
5d2−4d1−d1
which leads to a better bound than 16dminLemma 3.1. But we are satisfied withLemma 3.1.
A main tool in provingTheorem 2.4is Euler’s summation formula in the form
n
i1
fi
n
1
fxdxf1 fn
2
r
j1
B2j
2j!
f2j−1n−f2j−11 Rr, 3.4
wherer ∈ Nis a suitable chosen parameter, and the remainderRr is defined by a periodic
Bernoulli polynomialP2r1x, namely
Rr 1
2r1!
n
1
P2r1xf2r1xdx, 3.5
with
P2r1x −1r−12r1!
∞
j1
2 sin2πjx
2πj2r1 . 3.6
Applying the summation formula to the functionfx 1/x, we getsee8, equation5
n−1
i1
1
i logn
1
2−
1
2n
r
j1
B2j
2j
1− 1
n2j
− n
1
P2r1x
x2r2 dx, n, r ∈N. 3.7
It follows that
n2−1
in
1
i −logn 1 2n−
1 2n2
r
j1
B2j
2j
1 n2j −
1 n4j
−
n2
n
P2r1x
x2r2 dx, n, r ∈N. 3.8
We proveTheorem 2.4fora≥b. The casea < bis treated similarly. So we have again by the
above summation formula that
an−1
ibn
1 i −log
a
b
1 b−
1 a
1
2n
r
j1
B2j
2jn2j
1 b2j −
1 a2j
− an
bn
P2r1x
x2r2 dx, n, r∈N.
First, we estimate the integral on the right-hand side of3.8. We have
n2
n
P2r1x
x2r2 dx
≤
n2
n
|P2r1x|
x2r2 dx≤
∞
n
|P2r1x|
x2r2 dx
≤ 22r1!
∞
n
1 x2r2
∞
j1
1
2πj2r1dx
22r1! 2π2r1
− 1
2r1x2r1
∞
xn
∞
j1
1 j2r1
22r!
2π2r1n2r1 ζ2r1<
32r!
2π2r1n2r1,
3.10
since 2ζ2r1 ≤ 2ζ3 < 3. Next, we assume thatn ≥ a. Hence bn, an ⊆ n, n2, and
therefore we estimate the integral on the right-hand side in3.9by
an
bn
P2r1x
x2r2 dx
≤
an
bn
|P2r1x|
x2r2 dx
≤ n2
n
|P2r1x|
x2r2 dx≤
32r!
2π2r1n2r1.
3.11
In the sequel we putr dm. Moreover, in the above formula we now replacenbydmk
with 0≤k≤dm. In order to estimate2r! we use Stirling’s formula
√
2πm
m e
m
< m!<
2πm1
m e
m
, m >0. 3.12
Then, it follows that
dmk2
dmk
P2r1x
x2r2 dx
≤
32r!
2π2r1dmk2r1 ≤
32r!
2π2r1dm2r1
32dm! 2π2dm1dm2dm1
≤ 3
π2dm1
2π dm2dm1 ·
2dm
e
2dm
≤ 3
√
3πdm
2πdmπe2dm,
and similarly we have
admk
bdmk
P2r1x
x2r2 dx
≤
3√3πdm
2πdmπe2dm, dm≥a. 3.14
By using the definition ofSninTheorem 2.4, the formula1.1forsn, and the identities3.8,
3.9, it follows that
Sn−γ−loga
b
sn−γ
sn−s2n san−sbn− 1
2n2
r
j1
B2j
2j
1 n2j
1 a2j −
1 b2j 1
− 1
n4j
sn−γ 1 2n 1 b− 1 a−1
n2
n
P2r1x
x2r2 dx−
an
bn
P2r1x
x2r2 dx,
3.15
wherer is specified tor dmandnton dmk. Moreover, we know from4, Lemma 2
that
dm
k0
−1dmkgk 1, m≥1. 3.16
By settingndmk, the above formula for the series transformation ofSdmksimplifies to
dm
k0
−1dmkgkSdmk−γ−log
a b dm
k0
−1dmkgksdmk−γ 1 2 1 b− 1 a−1
dm
k0
−1dmkgk
dmk
dm
k0
−1dmkgk
dmk2
dmk
P2r1x
x2r2 dx
−dm k0
−1dmkgk
admk
bdmk
P2r1x
x2r2 dx
≤dm k0
−1dmkgksdmk−γ
dm
k0
gk
dmk2
dmk
P2r1x
x2r2 dx
dm
k0
gk
admk
bdmk
P2r1x
x2r2 dx
< Cd·
1−1/dd
d−14d
m−2 dm
k0
gk 3
√
3πdm
πdmπe2dm,
wheredm ≥a, m≥2, andd ≥3. Here, we have used the results fromCorollary 1.7,3.13,
and3.14. The sum
dm
k0
−1dmkgk
dmk 3.18
vanishes, since for every real numberx >−dmwe have
dm
k0
−1dmk2d−1mk−1
dm
dm k
dmkx
1−d−1mx· · ·mx
dmxdm1 , 3.19
where on the right-hand side for an integerxwith−m ≤ x ≤ d−1m−1 one term in the
numerator equals to zero. The inequality
64
πe2
d
< 1−1/d
d
2d−1 3.20
holds for all integersd≥42. Now, usingLemma 3.1, we estimate the right-hand side in3.17
fordm≥aandd≥42 as follows:
dm
k0
−1dmkgkSdmk−γ−log
a b
< Cd·
1−1/dd
d−14d
m−2 dm
k0
3√3πdm
πdm
16dm
πe2dm
Cd·
1−1/dd
d−14d
m−2
3dm1
√
3dm
dm√π
1 4dm
64
πe2
dm
3.20
< Cd·
1−1/dd
d−14d
m−2
3dm1
√
3dm
dm2m√π
1−1/dd
d−14d m
≤Cd·
1−1/dd
d−14d
m−2 85
28
3d π
1−1/dd
d−14d m
≤c4
1−1/dd
d−14d m
.
3.21
The last but one estimate holds for all integersm≥2,d≥42, andc4is a suitable positive real
4. On the Denominators of
S
nIn this section we will investigate the size of the denominators bm of our series
transformations
am
bm dm
k0
−1dmkgkSkdm, 4.1
formtending to infinity, wheream∈Zandbm∈Nare coprime integers.
Theorem 4.1. For everym≥1 there is an integerZmwithZm>0,bm|Zm, and
logZm∼12d2m2, m−→ ∞. 4.2
Proof. We will need some basic facts on the arithmetical functionsϑxandψx. Let
ϑx
p≤x
logp, x >1,
ψx
p≤x
logx
logp
logp, x >1,
4.3
wherepis restricted on primes. Moreover, letDn: l.c.m1,2, . . . , nfor positive integersn.
Then,
ψn logDn, n≥1, 4.4
ψx∼ ϑx∼ πxlogx ∼ x, x−→ ∞, 4.5
where4.5follows from5, Theorem 420and the prime number theorem. By5, Theorem
118 von Staudt’s theoremwe know how to obtain the prime divisors of the denominators
of Bernoulli numbers B2k: The denominators of B2k are squarefree, and they are divisible
exactly by those primespwithp−1|2k. Hence,
B2k
p≤2k1
p ∈ Z, k1,2, . . .. 4.6
Next, let max{a, b} ≤dm≤n≤2dmnkdmare the subscripts ofSkdminTheorem 2.4.
First, we consider the following terms from the series transformation inSm:
an−1
j1
1 j −
bn−1
j1
1 j 2
n−1
j1
1 j −
n2−1
j1
1 j :
n2−1
j1
ej
with
ej : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1, if 1≤j≤n−1
−1, ifn≤j≤bn−1
0, if bn≤j≤an−1
−1, ifan≤j≤n2−1
a≥b,
ej : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1, if 1≤j≤n−1
−1, ifn≤j≤an−1
−2, ifan≤j≤bn−1
−1, ifbn≤j ≤n2−1.
a < b.
4.8
For everym≥1 there is a rationalxm/ymdefined by
xm
ym
dm
k0
−1dmkgk
kdm2−1
j1
ej
j , 4.9
wherexm∈Z,ym∈N,xm, ym 1, and
ym|Ym :D4d2m2, dm≥max{a, b}. 4.10
Similarly, we define rationalsum/vmby
um
vm dm
k0
−1dmkgk
× ⎛
⎝− 1
2kdm2
dm
j1
B2j
2j
1
kdm2j
1 a2j −
1 b2j 1
− 1
kdm4j
⎞ ⎠,
4.11
whereum∈Z,vm∈Nandum, vm 1. We have
kdm2j|kdm4dm, 0≤k≤dm, 1≤j ≤dm. 4.12
Therefore, using the conclusion4.6from von Staudt’s theorem, we get
vm|Vm :2ab2dmDdm ⎛
⎝
p≤2dm1
p
⎞
⎠D2dm4dm, dm≥max{a, b}. 4.13
Note thatD2dm l.c.m.dm, . . . ,2dm, since every integern1with 1 ≤ n1 < dmdivides at
From4.10and4.13we conclude on
bm|Zm:2ab2dmDdmD4d2m2D2dm4dm
⎛
⎝
p≤2dm1
p
⎞
⎠. 4.14
Hence we have from4.4and4.5that
logZmlog 22dmlogab ψdm ψ
4d2m2 4dmψ2dm ϑ2dm1
∼ log 22dmlogab dm4d2m28d2m2 2dm1
1log 232 logabdm12d2m2
∼12d2m2 m−→ ∞.
4.15
The theorem is proved.
Remark 4.2. On the one side we have shown that logYm∼4d2m2and logVm∼8d2m2. On the
other side, every primepdividingVmsatisfiesp≤max{a, b, dm,2dm1,2dm}2dm1 and
thereforepdividesYm D4d2m2. Conversely, all primespwith 2dm1 < p <4d2m2divide
Ym, but notVm. That means:Vmis much bigger thanYm, butVm is formed by powers of small
primes, whereasYm is divisible by many big primes.
5. Simplification of the Transformed Series
Let
Rn:−
1 2n2
dm
j1
B2j
2j
1 n2j
1 a2j −
1 b2j 1
− 1
n4j
, 5.1
such that
Sn an−1
j1
1 j −
bn−1
j1
1 j 2
n−1
j1
1 j −
n2−1
j1
1
j Rn. 5.2
In Theorem 2.4 the sequence Sn is transformed. In view of a simplified process we now
investigate the transformation of the series Sn − Rn. Therefore we have to estimate the
contribution ofRkdmto the series transformation inTheorem 2.4. For this purpose, we define
Em: dm
k0
−1dmk
2d−1mk−1 dm
dm
k
Rkdm −1
2
dm
k0
−1dmkgk
dmk2
dm
j1
B2j
2j
1 a2j −
1 b2j 1
dm
k0
−1dmkgk
dmk2j −
dm
k0
−1dmkgk
dmk4j
.
5.3
Lemma 5.1. For positive integersd, jandmone has
dm
k0
−1dmkgk
dmk2j −
−1dm
dm!2j−1!
1
0
umlogu2j−1 ∂
dm
∂udm
u2d−1m−11−udm du. 5.4
Proof. For integersk, rand a real numberρwithkρ >0 the identity
1
kρr 1
r−1!
∞
0
e−kρttr−1 dt 5.5
holds, which we apply with r 2j and ρ dm to substitute the fraction 1/dmk2j.
Introducing the new variableu:e−t, we then get
2m
k0
−1dmkgk
kdm2j −
−1dm
2j−1!
dm
k0
−1kgk
1
0
ukdm−1logu2j−1 du
−−1dm
2j−1!
1
0
dm
k0
−1kgkukd−1m−1
umlogu2j−1 du.
5.6
The sum inside the brackets of the integrand can be expressed by using the equation
n
k0 −1k
nτk
n
n
k
uτk ∂
n
∂un
unτ1−un
n!
, n, τ∈N∪ {0}, 5.7
in which we putndmandτ d−1m−1. This gives the identity stated in the lemma.
The following result deals with the casej1, in which we express the finite sum by a double
integral on a rational function.
Corollary 5.2. For every positive integermone has
dm
k0
−1dmkgk
dmk2 −1
d−1m 1
0
1
0
1−udm1−wmu2d−1m−1wd−1m−1
1−1−uwdm1 du dw. 5.8
Proof. Setj 1 inLemma 5.1, and note that
logu−1−u
1
0
dw
Hence,
dm
k0
−1dmkgk
dmk2 −
−1dm
dm!
1
0
umlogu ∂
dm
∂udm
u2d−1m−11−udm du
−1dm
dm!
1
0
1
0
1−uum
1−1−uw ∂dm
∂udm
u2d−1m−11−udm du dw.
5.10
Let s be any positive integer. Then we have the following decomposition of a rational
function, in whichuis considered as variable andwas parameter:
us
1−1−uw
s−1
ν0
w−1ν wν1 u
s−ν−1
w−1 w
s 1
1−1−uw. 5.11
We additionally assume thats−1 < dm. Then, differentiating this identitydm-times with
respect tou, the polynomial inuon the right-hand side vanishes identically:
∂dm
∂udm
us
1−1−uw
w−1 w
s −1dmdm! wdm
1−1−uwdm1. 5.12
Therefore, we get from5.10by iterated integrations by parts:
dm
k0
−1dmkgk
dmk2 1
dm!
1
0
1
0
u2d−1m−11−udm ∂
dm
∂udm
um−um1 1−1−uw
du dw
1
dm!
1
0
1
0
u2d−1m−11−udm
w−1
w
m −
w−1 w
m1
× −1dmdm!wdm du dw
1−1−uwdm1 .
5.13
The corollary is proved by noting that
w−1 w
m −
w−1 w
m1
−1m 1−w
m
6. Estimating
E
mIn this section we estimateEm defined in5.3. Substituting 1−uforuinto the integral in
Lemma 5.1and applying iterated integration by parts, we get
dm
k0
−1dmkgk
dmk2j
− −1dm
dm!2j−1!
1
0
∂dm
∂udm
1−umlog1−u2j−1 1−u2d−1m−1udm du.
6.1
Set
fu: 1−umlog1−u2j−1, 6.2
wheremandjare kept fixed. We havef0 0. For an integerk >0 we use Cauchy’s formula
fka k! 2πi
C
fz
z−ak1 dz 6.3
to estimate |fk0|. LetC denote the circle in the complex plane centered around 0 with
radiusR:1−1/2k. Witha0 andfzdefined above, Cauchy’s formula yields the identity
fk0 k! 2πRk
π
−π
e−ikφ1−Reiφ mlog2j−11−Reiφ dφ. 6.4
For the complex logarithm function occurring in6.4we cut the complex plane along the
negative real axis and exclude the origin by a small circle. All argumentsφ of a complex
numberz /∈−∞,0are taken from the interval−π, π. Therefore, using 1−Reiφ1−Rcosφ− iRsinφ, we get
1−Reiφ 1R2−2Rcosφ:AR, φ,
arg1−Reiφ −arctan
Rsinφ
1−Rcosφ
.
6.5
Hence,
log1−Reiφ ln1R2−2Rcosφ−iarctan
Rsinφ
1−Rcosφ
1
2ln
AR, φ−iarctan
Rsinφ
1−Rcosφ
.
Thus, it follows from6.4that
fk0≤ k!
2πRk π
−π
1−Reiφm·log1−Reiφ 2j−1 dφ
k!
2πRk π
−π
AR, φm/2
1 4ln
2AR, φarctan2
Rsinφ
1−Rcosφ
2j−1/2
dφ
k!
πRk π
0
AR, φm/2
1 4ln
2AR, φarctan2
Rsinφ
1−Rcosφ
2j−1/2
dφ.
6.7
From 0< R <1 we conclude on
0<1−R21R2−2R≤1R2−2RcosφAR, φ<4, 0≤φ≤π,
0≤ Rsinφ
1−Rcosφ ≤
sinφ 1−cosφ,
0< φ≤π.
6.8
Since arctan is a strictly increasing function, we get
arctan
Rsinφ
1−Rcosφ
≤arctan
sinφ
1−cosφ
arctan cot
φ
2
arctan
tan
π−φ
2
π−φ
2 ,
0< φ≤π.
6.9
For 0< R <1, this upper bound also holds forφ0. Finally, we note thatRk 1−1/2kk≥
1/2. Altogether, we conclude from6.7on
fk0≤ k!
πRk π
0
4m/2
ln24
4 arctan
2
sinφ
1−cosφ
2j−1/2
dφ
≤ 2m1k!
π
π
0
ln22
π−φ
2
2 2j−1/2
dφ
≤ 2m1k!
π
π
0
ln22π
2
4
j−1/2
dφ
≤ 2m1k!
π
π
0
3j−1/2 dφ ≤ 2m13jk!.
It follows that the Taylor series expansion offu,
fu
∞
k0
fk0
k! u
k, 6.11
converges at least for−1< u <1. Then,
fdmu
∞
kdm
fk0
k−dm! u
k−dm∞ k0
fkdm0
k! u
k, 6.12
and the estimate given by6.10implies for 0< u <1 that
fdmu≤
∞
k0
fkdm0
k! u
k ≤ 2m13j ∞ k0
kdm!
k! u
k
2m13jdm!∞ k0
kdm
k
uk 2m13jdm!
1−udm1 .
6.13
Combining6.13with the result from6.1, we get form >1
dm
k0
−1dmkgk
dmk2j
≤
2m13j
2j−1!
1
0
1−ud−1m−2udm du
2m13j
2j−1!
Γdm1Γd−1m−1
Γ2d−1m
2d−1
d−1 ·
2m13j
2j−1!d−1m−1 · 1
2d−1m
dm
.
6.14
We estimate the binomial coefficient by Stirling’s formula 3.12. For this purpose we
additionally assume thatm≥2d−1:
2d−1m dm
≥
2d−1m
2πdm1d−1m1
2d−12d−1 ddd−1d−1
m
≥
2d−1 2πd2m
2d−12d−1 ddd−1d−1
m
.
6.15
We now assumem≥2d−1 and substitute the above inequality into6.14:
dm
k0
−1dmkgk
dmk2j
≤
d2d−12m13j√2πm
2j−1!d−1d−1m−1√2d−1
ddd−1d−1 2d−12d−1
m
For all integersm≥1 andd≥1 we have
2d−1√2πm
√
2d−1 <2
πdm, d−1d−1m−1≥d−1d−2m. 6.17
Thus we have proven the following result.
Lemma 6.1. For all integersd, mwithd≥3 andm≥2d−1 one has
dm
k0
−1dmkgk
dmk2j
<
2m23j√πd3
2j−1!d−1d−2√m
ddd−1d−1 2d−12d−1
m
. 6.18
Next, we need an upper bound for the Bernoulli numbersB2jcf.9, 23.1.15:
B2j≤ 22j!
2π2j1−21−2j ≤
42j!
2π2j,
j≥1. 6.19
Letd≥3 andm≥max{2d−1, a/2}. Using this andLemma 6.1, we estimateEmin5.3:
|Em|<
3 2
√
πd3 2m2 d−1d−2√m
ddd−1d−1 2d−12d−1
m
√
πd3 2m2 d−1d−2√m
ddd−1d−1 2d−12d−1
m
×dm j1
B2j
2j
a12j −
1 b2j 1
3j
2j−1! 32j
4j−1!
≤ √
πd3 2m2 d−1d−2√m
ddd−1d−1 2d−12d−1
m
× ⎛ ⎝3
2
dm
j1
42j−1!
2π2j
2·3j
2j−1! 32j
4j−1!
⎞ ⎠
< 4
√
πd3 d−1d−2√m
2ddd−1d−1 2d−12d−1
m⎛ ⎝3
2 8
∞
j1
⎛ ⎝
√
3 2π
2j
3 2π
2j⎞ ⎠
⎞ ⎠
< 19
√
πd3 d−1d−2√m
2ddd−1d−1 2d−12d−1
m
.
Now, let
Tn: an−1
j1
1 j −
bn−1
j1
1 j 2
n−1
j1
1 j −
n2−1
j1
1
j
n2−1
j1
ej
j , n >1, 6.21
with the numbersejintroduced in the proof ofTheorem 4.1. By definition ofRnandSnwe
then haveTn Sn−Rn, and therefore we can estimate the series transformation of Tn by
applying the results fromTheorem 2.4and6.20. Again, letm≥max{2d−1, a/2}andd≥42.
dm
k0
−1dmk
⎛
⎝2d−1mk−1
dm
⎞ ⎠
⎛ ⎝dm
k
⎞
⎠Tkdm−γ−loga
b
≤dm k0
−1dmkgkSkdm−γ−log
a b
dm
k0
−1dmkgkRkdm
dm
k0
−1dmkgkSkdm−γ−log
a b
|Em|
< c4·
1−1/dd
d−14d m
19
√
πd3 d−1d−2√m
2ddd−1d−1 2d−12d−1
m
.
6.22
By similar arguments we get the same bound whenb > a. Ford≥3 it can easily be seen that
2ddd−1d−1 2d−12d−1
22d−1
d−1 ·
1−1/dd
1−1/2d2d · 1 4d <
18
4d1. 6.23
Thus, we finally have proven the following theorem.
Theorem 6.2. Let
Tn: an−1
j1
1 j −
bn−1
j1
1 j 2
n−1
j1
1 j −
n2−1
j1
1
j, n >1, 6.24
where a, b are positive integers. Let d ≥ 42 be an integer. Then, there is a positive constant c5 depending at most ona, banddsuch that
dm
k0 −1k
2d−1mk−1 dm
dm
k
Tkdm−γ−loga
b
< √cm5
18 4d1
m
, m≥1.
7. Concluding Remarks
It seems that inTheorem 6.2a smaller bound holds.
Conjecture 7.1. Let a,b be positive integers. Letd≥2 be an integer. Then there is a positive constant c6depending at most ona, banddsuch that for all integersm≥1 one has
dm
k0
−1dmk
2d−1mk−1 dm
dm
k
Tkdm−γ−loga
b
< c6·
1−1/dd
d−14d m
.
7.1
A proof of this conjecture would be implied by suitable bounds for the integral stated inLemma 5.1. Forj1 such a bound follows from the double integral given inCorollary 5.2:
dm
k0
−1dmkgk
dmk2
1
0
1
0
1−udm1−wmu2d−1m−1wd−1m−1
1−1−uwdm1 du dw
1
0
1
0
1−u21−wu2
1−1−uw3
1−uu2w
1−1−uw
d−2
×
1−ud1−wu2d−1wd−1 1−1−uwd
m−1
du dw
≤ 1
4d−2
1−1/dd
d−14d
m−11
0
1
0
1−u21−wu2
1−1−uw3 du dw
2d−1
31−1/dd ·
1−1/dd
d−14d m
, m≥1,
7.2
where the double integral in the last but one line equals to 1/24.
Note that the rational functions
1−uu2w 1−1−uw,
1−ud1−wu2d−1wd−1
1−1−uwd , 7.3
take their maximum values 42−dand1−1/dd/d−14dinside the unit square0,1×0,1
atu, w 1/2,1andu, w 1/2,2d−2/2d−1, respectively. Finally, we compare
the bound for the series transformation given by Theorem 2.4with the bound proven for
Theorem 6.2. InTheorem 2.4the bound is
T1d, m:c4·
1−1/dd
d−14d m
whereas we have inTheorem 6.2that
T2d, m: √c5 m
18 4d1
m
, d≥42, m≥1. 7.5
For fixedd≥42 and sufficiently largemit is clear on the one hand thatT1d, m< T2d, m, but on the other hand we have
lim
m→ ∞dlim→ ∞
logT1d, m
logT2d, m 1dlim→ ∞mlim→ ∞
logT1d, m
logT2d, m. 7.6
Conversely, fordtending to infinity, one gets
−log|T1d, m| dmlog 4, −log|T2d, m| dmlog 4, 7.7
with implicit constants depending at most onm. For the denominatorsbmof the transformed
series Skdm in Theorem 2.4we have the bound logbm d2m2 from Theorem 4.1, and a
similar inequality holds for the denominators of the transformed seriesTkdminTheorem 6.2.
References
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2 T. Rivoal, “Polyn ˆomes de type Legendre et approximations de la constante d’Euler,” notes, 2005,
http://www-fourier.ujf-grenoble.fr/∼rivoal/articles/euler.pdf.
3 K. H. Pilehrood and T. H. Pilehrood, “Arithmetical properties of some series with logarithmic coefficients,” Mathematische Zeitschrift, vol. 255, no. 1, pp. 117–131, 2007.
4 C. Elsner, “On a sequence transformation with integral coefficients for Euler’s constant. II,” Journal of Number Theory, vol. 124, no. 2, pp. 442–453, 2007.
5 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, UK, 1984.
6 A. I. Aptekarev, Ed., Rational Approximation of Euler’s Constant and Recurrence Relations, Sovremennye Problemy Matematiki, Vol. 9, MIANSteklov Institute, Moscow, Russia, 2007.
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http://www-fourier.ujf-grenoble.fr/∼rivoal/articles/eulerconstant.pdf.
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