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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,carsten.elsner@fhdw.de

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|γlogqam/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

(2)

Proposition 1.1see1. For any integersn1 andτ2 one has

n

k0 −1nk

nkτ−1 n

n

k

·skτγ

τ−1!

2nn1n2· · ·nτ. 1.3

Particularly, by choosingτn2, one gets the following result.

Corollary 1.2. For any integern2one has

n

k0 −1nk

2nk−1 n

n

k

·snkγ

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. Letn1,τ1≥1,andτ2 ≥1 be integers. Additionally one assumes that

(3)

Then one has

n

k0 −1nk

1k

n

n

k

·skτ2−γ

−1n1

1

0

1

1−u

1 log u

·2−τ1−1·

n

∂un

unτ11un n!

du.

1.8

Proposition 1.6see4. Letn1,τ1≥1 andτ2≥1 be integers. Additionally one assumes that

1τ1 ≤ τ2 ≤ 11. 1.9

Then one has

n

k0 −1nk

1k

n

n

k

·skτ2−γ

−12−τ1

1

0

1

0

wt·1−u

1un1tτ2−τ1−1t1−τ21

1−utn1 du dt,

1.10

with

wt: 1

t·

π2log2

1 t −1

. 1.11

Setting

2dm, τ1 d−1m−1, d≥2, 1.12

one gets an explicit upper bound fromProposition 1.6

Corollary 1.7. For integersm2,d3, one has

dm

k0

−1dmk

2d−1mk−1 dm

dm

k

·skdmγ < 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

·sk2mγ <

16 7π

2

· 1

(4)

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:

km

Λkm. 1.16

Then, forε: log 55/4−1>0.0018, there is some integerm0such that

Bmeψ4m < e41εm55m mm0. 1.17

Multiplying1.14byBm, we deduce the following corollary.

Corollary 1.8. There is an integerm0such that one has for all integersmm0that

Bm

2m

k0 −1k

3mk−1

2m

2m

k

·logk2m γBmAm <

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

γpn

qnc0e−2

2n, q nc1

n1/4 2n!

n! e

2n, 2.2

(5)

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

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

−3n36n27n13 U

n1 n23Un,

2.6

provides two sequences of rational numberspnn≥0andqnn≥0withp0 −1,p1 11,p271 and

(6)

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. Leta1,b1,d42 andm1 be positive integers, and

Sn: an−1

j1

1 j

bn−1

j1

1 j 2

n−1

j1

1 j

n21

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

Skdmγ−loga

b

< c

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≤kdm. 3.1

Proof. Applying the well known inequalityg

h ≤2

g, we get

2d−1mk−1 dm

dm

k

≤22d−1mdm−12dm24dmm−1< 16dm. 3.2

This proves the lemma.

gktakes its maximum value forkk0with

k0

5d24d1d1

(7)

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−1nf2j−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

n21

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 2.4forab. 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.

(8)

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 thatna. Hence bn, ann, 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≤kdm. 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,

(9)

and similarly we have

admk

bdmk

P2r1x

x2r2 dx

3√3πdm

2πdmπe2dm, dma. 3.14

By using the definition ofSninTheorem 2.4, the formula1.1forsn, and the identities3.8,

3.9, it follows that

Snγ−loga

b

snγ

sns2n sansbn− 1

2n2

r

j1

B2j

2j

1 n2j

1 a2j

1 b2j 1

− 1

n4j

snγ 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

−1dmkgkSdmkγ−log

a b dm

k0

−1dmkgksdmkγ 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

−1dmkgksdmkγ

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,

(10)

wheredma, 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−mxd−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

fordmaandd≥42 as follows:

dm

k0

−1dmkgkSdmkγ−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

dmπ

1 4dm

64

πe2

dm

3.20

< Cd·

1−1/dd

d−14d

m−2

3dm1

3dm

dm2mπ

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

(11)

4. On the Denominators of

S

n

In 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 everym1 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

px

logp, x >1,

ψx

px

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

ψxϑxπxlogxx, 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} ≤dmn≤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

n21

j1

1 j :

n21

j1

ej

(12)

with

ej : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1, if 1≤jn−1

−1, ifnjbn−1

0, if bnjan−1

−1, ifanjn21

ab,

ej : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1, if 1≤jn−1

−1, ifnjan−1

−2, ifanjbn−1

−1, ifbnjn21.

a < b.

4.8

For everym≥1 there is a rationalxm/ymdefined by

xm

ym

dm

k0

−1dmkgk

kdm21

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≤kdm, 1≤jdm. 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

(13)

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

n21

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 SnRn. 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

(14)

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

ekρttr−1 dt 5.5

holds, which we apply with r 2j and ρ dm to substitute the fraction 1/dmk2j.

Introducing the new variableu:et, 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

unτ1un

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

(15)

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 1 u

sν−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

umum1 1−1−uw

du dw

1

dm!

1

0

1

0

u2d−1m−11−udm

w1

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

(16)

6. Estimating

E

m

In 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

zak1 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

eikφ1−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−RcosφiRsinφ, we get

1−Reiφ 1R22Rcosφ:AR, φ,

arg1−Reiφ −arctan

Rsinφ

1−Rcosφ

.

6.5

Hence,

log1−Reiφ ln1R22Rcosφiarctan

Rsinφ

1−Rcosφ

1

2ln

AR, φiarctan

Rsinφ

1−Rcosφ

.

(17)

Thus, it follows from6.4that

fk0≤ k!

2πRk π

π

1−Reiφm·log1−Reiφ 2j−1

k!

2πRk π

π

AR, φm/2

1 4ln

2AR, φarctan2

Rsinφ

1−Rcosφ

2j−1/2

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≤φπ,

0≤ Rsinφ

1−Rcosφ

sinφ 1−cosφ,

0< φπ.

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

πφ

2

πφ

2 ,

0< φπ.

6.9

For 0< R <1, this upper bound also holds forφ0. Finally, we note thatRk 11/2kk

1/2. Altogether, we conclude from6.7on

fk0≤ k!

πRk π

0

4m/2

ln24

4 arctan

2

sinφ

1−cosφ

2j−1/2

≤ 2m1k!

π

π

0

ln22

πφ

2

2 2j−1/2

≤ 2m1k!

π

π

0

ln22π

2

4

j−1/2

≤ 2m1k!

π

π

0

3j−1/2 2m13jk!.

(18)

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

kdm! u

kdmk0

fkdm0

k! u

k, 6.12

and the estimate given by6.10implies for 0< u <1 that

fdmu

k0

fkdm0

k! u

k 2m13jk0

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!

Γdmd−1m−1

Γ2d−1m

2d−1

d−1 ·

2m13j

2j−1!d−1m−1 · 1

2d1m

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 ddd1d−1

m

2d−1 2πd2m

2d−12d−1 ddd1d−1

m

.

6.15

We now assumem≥2d−1 and substitute the above inequality into6.14:

dm

k0

−1dmkgk

dmk2j

d2d−12m13j2πm

2j−1!d−1d−1m−1√2d−1

ddd1d−1 2d−12d−1

m

(19)

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, mwithd3 andm≥2d1 one has

dm

k0

−1dmkgk

dmk2j

<

2m23jπd3

2j−1!d−1d−2√m

ddd1d−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

ddd1d−1 2d−12d−1

m

πd3 2m2 d−1d−2√m

ddd1d−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

ddd1d−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

2ddd1d−1 2d−12d−1

m⎛ ⎝3

2 8

j1

⎛ ⎝

3 2π

2j

3 2π

2j⎞ ⎠

⎞ ⎠

< 19

πd3 d−1d−2√m

2ddd1d−1 2d−12d−1

m

.

(20)

Now, let

Tn: an−1

j1

1 j

bn−1

j1

1 j 2

n−1

j1

1 j

n21

j1

1

j

n21

j1

ej

j , n >1, 6.21

with the numbersejintroduced in the proof ofTheorem 4.1. By definition ofRnandSnwe

then haveTn SnRn, 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

Tkdmγloga

b

dm k0

−1dmkgkSkdmγ−log

a b

dm

k0

−1dmkgkRkdm

dm

k0

−1dmkgkSkdmγ−log

a b

|Em|

< c

1−1/dd

d−14d m

19

πd3 d−1d−2√m

2ddd1d−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

2ddd1d−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

n21

j1

1

j, n >1, 6.24

where a, b are positive integers. Let d42 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

Tkdmγ−loga

b

<cm5

18 4d1

m

, m≥1.

(21)

7. Concluding Remarks

It seems that inTheorem 6.2a smaller bound holds.

Conjecture 7.1. Let a,b be positive integers. Letd2 be an integer. Then there is a positive constant c6depending at most ona, banddsuch that for all integersm1 one has

dm

k0

−1dmk

2d−1mk−1 dm

dm

k

Tkdmγ−loga

b

< c

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−dand11/dd/d14dinside 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:c

1−1/dd

d−14d m

(22)

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

1 C. Elsner, “On a sequence transformation with integral coefficients for Euler’s constant,” Proceedings of the American Mathematical Society, vol. 123, no. 5, pp. 1537–1541, 1995.

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.

7 T. Rivoal, “Rational approximations for values of derivatives of the Gamma function,”

http://www-fourier.ujf-grenoble.fr/∼rivoal/articles/eulerconstant.pdf.

8 D. E. Knuth, “Euler’s constant to 1271 places,” Mathematics of Computation, vol. 16, no. 79, pp. 275–281, 1962.

References

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