On Rational Approximations to Euler’s Constant γ and to γ log a/b

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|γlogq−am/bm| 1−1/d^d/d−14^dmandbm|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

s_n: 1

1 1 2 1

3· · · 1 n−1

−logn n≥2. 1.1

It is well known that the sequencesn_n≥1converges to Euler’s constantγ0,577. . ., where s_nγO

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 seriessn_n≥1 with integer coefficients which improves the rate of convergence. Letτ be an additional positive integer parameter.

Proposition 1.1see1. For any integersn≥1 andτ ≥2 one has

n k0

−1^nk

nkτ−1 n

n k

·s_kτ−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

−1^nk

2nk−1 n

n k

·s_nk−γ

≤ 1 2n²_2n

n

≤ 1

n^3/2·4ⁿ. 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

−1^k

2n2k n

n k

s_2kn1

O 1

n27^n/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

−1^nk nk

n n

k

s_kn1 1

4^non. 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

1τ1 ≤ τ2. 1.7

Then one has

n k0

−1^nk

nτ₁k

n

n k

·s_kτ2−γ −1ⁿ¹

₁

0

1

1−u 1 log u

·u^{τ2−τ1−1}· ∂ⁿ

∂uⁿ

u^nτ11−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

−1^nk

nτ₁k

n

n k

·s_kτ2−γ −1^nτ2−τ1

₁

0

₁

0

wt·1−u^nτ1uⁿ1−t^{τ2−τ1−1}t^nτ1−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

−1^dmk

2d−1mk−1 dm

dm k

·s_kdm−γ < Cd·

1−1/d^d d−14^d

_m−2

, 1.13

where 0< Cd≤1/16π²is some constant depending only ond. Ford2 one gets

2m k0

−1^k

3mk−1 2m

2m k

·s_k2m−γ <

16 7π

2

· 1

64^m m≥1. 1.14

For an application ofCorollary 1.7let the integersBmandAmbe defined by

Bm:l.c.m.1,2,3, . . . ,4m, Am:Bm

2m k0

−1^k

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≤mΛk∼m. 1.16

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

Bme^ψ4m < e^41εm55^m m≥m0. 1.17

Multiplying1.14byB_m, we deduce the following corollary.

Corollary 1.8. There is an integerm0such that one has for all integersm≥m0that

Bm

2m k0

−1^k

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. Letpn_n≥0andqn_n≥0be two solutions of the linear recurrence

16n−15n1un1

128n³40n²−82n−45 un

−n

256n³−240n²64n−7 u_n−1nn−116n1un−2

2.1

withp₀0,p₁2,p₂31/2 andq₀1,q₁3,q₂25. Then, one hasq_n∈Z,Dnpn∈Z, and γ−p_n

q_n

∼c0e^−2√2n, qn∼ c1

n^1/4 2n!

n! e^√2n, 2.2

with two positive constantsc₀, c₁.

It seems interesting to replace the fractionpn/qnby A_n

Bn : Dnpn

Dnqn, 2.3

and to estimate the remainder in terms ofB_n.

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

logB_n/log logB_n

<

γ−An

B_n

< 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 8n27U_n1 n2²8n198n27Un,

2.5

provides two sequences of rational numberspn_n≥0andqn_n≥0withp0−1,p14,p277/4 and q01,q1 7,q265/2 such thatpn/qn_n≥0converges toγ.

(ii) The recurrence

n1n2n3U_n3

3n²19n29 n1U_n2

−

3n³6n²−7n−13 U_n1 n2³U_n,

2.6

provides two sequences of rational numberspn_n≥0andqn_n≥0withp0 −1,p1 11,p271 and q₀0,q₁ 8,q₂56 such thatpn/q_nn≥0converges to log2 γ.

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

S_n:^an−1

j1

1 j −^bn−1

j1

1 j 2

n−1

j1

1 j −ⁿ

2−1

j1

1 j − 1

2n^{2 dm}

j1

B_2j 2j

1 n^2j

1 a^2j − 1

b^2j 1

− 1 n^4j

, n≥1.

2.7

Then,

dm k0

−1^dmk

2d−1mk−1 dm

dm k

S_kdm−γ−loga b < c4·

1−1/d^d d−14^d

_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

<16^dm 0≤k≤dm. 3.1

Proof. Applying the well known inequality_g

h ≤2^g, we get 2d−1mk−1

dm

dm k

≤2^{2d−1mdm−1}2^dm2^4dm−m−1< 16^dm. 3.2

This proves the lemma.

gktakes its maximum value forkk₀with

k₀

√5d²−4d1−d1

2 mO1, 3.3

which leads to a better bound than 16^dminLemma 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

B_2j 2j

!

f^2j−1n−f^2j−11 R_r, 3.4

wherer ∈ Nis a suitable chosen parameter, and the remainderRr is defined by a periodic Bernoulli polynomialP_2r1x, namely

Rr 1 2r1!

_n

1

P_2r1xf^2r1xdx, 3.5

with

P_2r1x −1^r−12r1!^∞

j1

2 sin 2πjx

2πj_2r1 . 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 n^2j

− _n

1

P_2r1x

x^2r2 dx, n, r ∈N. 3.7

It follows that

n²−1 in

1

i −logn 1 2n− 1

2n^{2 r}

j1

B2j

2j 1

n^2j − 1 n^4j

− _n²

n

P_2r1x

x^2r2 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 −loga b

1 b− 1

a 1

2n^r

j1

B_2j 2jn^2j

1 b^2j − 1

a^2j

− _an

bn

P_2r1x

x^2r2 dx, n, r∈N.

3.9

First, we estimate the integral on the right-hand side of3.8. We have

_n²

n

P_2r1x x^2r2 dx

≤ _n²

n

|P_2r1x|

x^2r2 dx≤ _∞

n

|P_2r1x|

x^2r2 dx

≤ 22r1!

_∞

n

1 x^2r2

∞ j1

1

2πj_2r1dx 22r1!

2π^2r1

− 1

2r1x^2r1 _∞

xn

∞ j1

1 j^2r1

22r!

2π^2r1n^2r1 ζ2r1< 32r! 2π^2r1n^2r1,

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

P_2r1x x^2r2 dx

≤ _an

bn

|P2r1x|

x^2r2 dx

≤ _n²

n

|P_2r1x|

x^2r2 dx≤ 32r!

2π^2r1n^2r1.

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

P_2r1x x^2r2 dx

≤ 32r!

2π^2r1dmk^2r1 ≤ 32r!

2π^2r1dm^2r1

32dm!

2π^2dm1dm^2dm1

≤ 3

π2dm1 2π dm^2dm1 ·

2dm e

_2dm

≤ 3√ 3πdm 2πdmπe^2dm,

3.13

and similarly we have

_admk

bdmk

P_2r1x x^2r2 dx

≤ 3√ 3πdm

2πdmπe^2dm, dm≥a. 3.14

By using the definition ofS_ninTheorem 2.4, the formula1.1fors_n, and the identities3.8, 3.9, it follows that

Sn−γ−loga b

sn−γ

sn−s²_n san−sbn− 1 2n^{2 r}

j1

B2j

2j 1

n^2j 1

a^2j − 1 b^2j 1

− 1 n^4j

s_n−γ

1 2n

1 b− 1

a−1

_n²

n

P_2r1x x^2r2 dx−

_an

bn

P_2r1x x^2r2 dx,

3.15

wherer is specified tor dmandnton dmk. Moreover, we know from4, Lemma 2 that

dm k0

−1^dmkgk 1, m≥1. 3.16

By settingndmk, the above formula for the series transformation ofS_dmksimplifies to

dm k0

−1^dmkgkSdmk−γ−loga b

dm k0

−1^dmkgk

s_dmk−γ 1

2 1

b− 1 a−1

_dm

k0

−1^dmkgk dmk

^dm

k0

−1^dmkgk _dmk²

dmk

P_2r1x x^2r2 dx

−^dm

k0

−1^dmkgk

_admk

bdmk

P_2r1x x^2r2 dx

≤

dm k0

−1^dmkgk

s_dmk−γ ^dm

k0

gk

_dmk²

dmk

P_2r1x x^2r2 dx

^dm

k0

gk

_admk

bdmk

P_2r1x x^2r2 dx

< C_d·

1−1/d^d d−14^d

_m−2 ^dm

k0

gk 3√ 3πdm πdmπe^2dm,

3.17

wheredm ≥a, m≥2, andd ≥3. Here, we have used the results fromCorollary 1.7,3.13, and3.14. The sum

dm k0

−1^dmkgk

dmk 3.18

vanishes, since for every real numberx >−dmwe have

dm k0

−1^dmk_2d−1mk−1

dm

dm k

dmkx 1−d−1mx· · ·mx

dmx_dm1 , 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

−1^dmkgkS_dmk−γ−loga b

< Cd·

1−1/d^d d−14^d

_m−2 ^dm

k0

3√ 3πdm πdm

16^dm πe^2dm

Cd·

1−1/d^d d−14^d

_m−2

3dm1√ 3dm dm√

π

1 4^dm

64 πe²

_dm

3.20< C_d·

1−1/d^d d−14^d

_m−2

3dm1√ 3dm dm2^m√

π

1−1/d^d d−14^d

m

≤C_d·

1−1/d^d d−14^d

_m−2 85

28

3d π

1−1/d^d d−14^d

m

≤c₄

1−1/d^d d−14^d

m

. 3.21

The last but one estimate holds for all integersm≥2,d≥42, andc4is a suitable positive real constant depending ond. This completes the proof ofTheorem 2.4.

4. On the Denominators of S

In this section we will investigate the size of the denominators b_m of our series transformations

a_m b_m ^dm

k0

−1^dmkgkSkdm, 4.1

formtending to infinity, wherea_m∈Zandb_m∈Nare coprime integers.

Theorem 4.1. For everym≥1 there is an integerZ_mwithZ_m>0,b_m|Zm, and

logZ_m∼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 logD_n, 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 B_2k: The denominators of B_2k are squarefree, and they are divisible exactly by those primespwithp−1|2k. Hence,

B_2k

p≤2k1

p ∈ Z, k1,2, . . .. 4.6

Next, let max{a, b} ≤dm≤n≤2dmnkdmare the subscripts ofS_kdminTheorem 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 −ⁿ

2−1

j1

1 j :ⁿ

2−1

j1

e_j

j , 4.7

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,

e_j :

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

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 x_m

y_m ^dm

k0

−1^dmkgk^kdm

2−1

j1

e_j

j , 4.9

wherex_m∈Z,y_m∈N,xm, y_m 1, and

y_m|Y_m :D_4d²_m², dm≥max{a, b}. 4.10 Similarly, we define rationalsum/vmby

um

v_m ^dm

k0

−1^dmkgk

×

⎛

⎝− 1

2kdm^{2 dm}

j1

B2j

2j

1 kdm^2j

1 a^2j − 1

b^2j 1

− 1

kdm^4j

⎞

⎠,

4.11

whereu_m∈Z,v_m∈Nandum, v_m 1. We have kdm^2j|kdm^4dm,

0≤k≤dm, 1≤j ≤dm

. 4.12

Therefore, using the conclusion4.6from von Staudt’s theorem, we get

v_m|V_m :2ab^2dmD_dm

⎛

⎝

p≤2dm1

p

⎞

⎠D2dm^4dm, dm≥max{a, b}. 4.13

Note thatD2dm l.c.m.dm, . . . ,2dm, since every integern1with 1 ≤ n1 < dmdivides at least one integern₂withdm≤n₂≤2dm.

From4.10and4.13we conclude on

b_m|Z_m:2ab^2dmD_dmD_4d²_m²D2dm^4dm

⎛

⎝

p≤2dm1

p

⎞

⎠. 4.14

Hence we have from4.4and4.5that logZmlog 22dmlogab ψdm ψ

4d²m² 4dmψ2dm ϑ2dm1

∼ log 22dmlogab dm4d²m²8d²m² 2dm1 1log 2

32 logab

dm12d²m²

∼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 thereforepdividesY_m D_4d²_m². 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

R_n:− 1 2n^{2 dm}

j1

B2j

2j 1

n^2j 1

a^2j − 1 b^2j 1

− 1 n^4j

, 5.1

such that

S_n^an−1

j1

1 j −^bn−1

j1

1 j 2

n−1

j1

1 j −ⁿ

2−1

j1

1

j R_n. 5.2

In Theorem 2.4 the sequence S_n 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 ofR_kdmto the series transformation inTheorem 2.4. For this purpose, we define

Em:^dm

k0

−1^dmk

2d−1mk−1 dm

dm k

R_kdm −1 2

dm k0

−1^dmkgk dmk²

^dm

j1

B2j

2j

1 a^2j − 1

b^2j 1 dm

k0

−1^dmkgk dmk^2j −^dm

k0

−1^dmkgk dmk^4j

.

5.3

A major step in estimatingE_mis to express the sums on the right-hand side by integrals.

Lemma 5.1. For positive integersd, jandmone has

dm k0

−1^dmkgk

dmk^2j − −1^dm dm!

2j−1

! ₁

0

u^m

logu_2j−1 ∂^dm

∂u^dm

u^2d−1m−11−u^dm du. 5.4

Proof. For integersk, rand a real numberρwithkρ >0 the identity 1

kρ_r 1 r−1!

_∞

0

e^−kρtt^r−1 dt 5.5

holds, which we apply with r 2j and ρ dm to substitute the fraction 1/dmk^2j. Introducing the new variableu:e^−t, we then get

2m k0

−1^dmkgk

kdm^2j −−1^dm 2j−1

! dm k0

−1^kgk ₁

0

u^kdm−1

logu_2j−1 du

−−1^dm 2j−1

! ₁

0

_dm

k0

−1^kgku^kd−1m−1

u^mlogu^2j−1 du.

5.6

The sum inside the brackets of the integrand can be expressed by using the equation

n k0

−1^k

nτk

n

n k

u^τk ∂ⁿ

∂uⁿ

u^nτ1−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

−1^dmkgk

dmk² −1^d−1m ₁

0

₁

0

1−u^dm1−w^mu^2d−1m−1w^d−1m−1

1−1−uw^dm1 du dw. 5.8

Proof. Setj 1 inLemma 5.1, and note that

logu−1−u ₁

0

dw

1−1−uw. 5.9

Hence,

dm k0

−1^dmkgk

dmk² −−1^dm dm!

₁

0

u^mlogu ∂^dm

∂u^dm

u^2d−1m−11−u^dm du

−1^dm dm!

₁

0

₁

0

1−uu^m 1−1−uw

∂^dm

∂u^dm

u^2d−1m−11−u^dm 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:

u^s

1−1−uw ^s−1

ν0

w−1^ν

w^ν1 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

∂u^dm

u^s 1−1−uw

w−1 w

_s −1^dmdm! w^dm

1−1−uw^dm1. 5.12

Therefore, we get from5.10by iterated integrations by parts:

dm k0

−1^dmkgk dmk² 1

dm!

₁

0

₁

0

u^2d−1m−11−u^dm ∂^dm

∂u^dm

u^m−u^m1 1−1−uw

du dw

1 dm!

₁

0

₁

0

u^2d−1m−11−u^dmw−1 w

_m

−

w−1 w

_m1

× −1^dmdm!w^dm du dw 1−1−uw^dm1 .

5.13

The corollary is proved by noting that w−1

w _m

−

w−1 w

_m1

−1^m 1−w^m

w^m1 . 5.14

6. Estimating E

In this section we estimateE_m defined in5.3. Substituting 1−uforuinto the integral in Lemma 5.1and applying iterated integration by parts, we get

dm k0

−1^dmkgk dmk^2j

− −1^dm dm!

2j−1

! ₁

0

∂^dm

∂u^dm

1−u^m

log1−u_2j−1

1−u^2d−1m−1u^dm du.

6.1

Set

fu: 1−u^m

log1−u_2j−1

, 6.2

wheremandjare kept fixed. We havef0 0. For an integerk >0 we use Cauchy’s formula

f^ka k!

2πi

C

fz

z−a^k1 dz 6.3

to estimate |f^k0|. 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

f^k0 k!

2πR^k _π

−πe^−ikφ

1−Re^{iφ m}log^2j−1

1−Re^iφ 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−Re^iφ1−Rcosφ− iRsinφ, we get

1−Re^iφ

1R²−2Rcosφ: A

R, φ ,

arg

1−Re^iφ −arctan

Rsinφ 1−Rcosφ

.

6.5

Hence,

log

1−Re^iφ ln

1R²−2Rcosφ−iarctan

Rsinφ 1−Rcosφ

1 2ln

A R, φ

−iarctan

Rsinφ 1−Rcosφ

.

6.6

Thus, it follows from6.4that f^k0≤ k!

2πR^k _π

−π

1−Re^iφm·log

1−Re^{iφ 2j−1} dφ

k!

2πR^k _π

−π

A

R, φm/2 1 4ln²

AR, φ

arctan²

Rsinφ 1−Rcosφ

^2j−1/2 dφ

k!

πR^k _π

0

A

R, φ_m/2 1 4ln²

AR, φ

arctan²

Rsinφ 1−Rcosφ

2j−1/2

dφ.

6.7

From 0< R <1 we conclude on

0<1−R²1R²−2R≤1R²−2RcosφA R, 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 thatR^k 1−1/2k^k≥ 1/2. Altogether, we conclude from6.7on

f^k0≤ k!

πR^k _π

0

4^m/2 ln²4

4 arctan²

sinφ 1−cosφ

2j−1/2

dφ

≤ 2^m1k!

π _π

0

ln²2

π−φ 2

2 _2j−1/2 dφ

≤ 2^m1k!

π _π

0

ln²2π² 4

_j−1/2 dφ

≤ 2^m1k!

π _π

0

3^j−1/2 dφ ≤ 2^m13^jk!.

6.10

It follows that the Taylor series expansion offu, fu ^∞

k0

f^k0

k! u^k, 6.11

converges at least for−1< u <1. Then,

f^dmu ^∞

kdm

f^k0

k−dm! u^k−dm∞

k0

f^kdm0

k! u^k, 6.12

and the estimate given by6.10implies for 0< u <1 that f^dmu≤^∞

k0

f^kdm0

k! u^k ≤ 2^m13^j ∞ k0

kdm!

k! u^k

2^m13^jdm!^∞

k0

kdm

k

u^k 2^m13^jdm!

1−u^dm1 .

6.13

Combining6.13with the result from6.1, we get form >1

dm k0

−1^dmkgk dmk^2j

≤ 2^m13^j 2j−1

! ₁

0

1−u^d−1m−2u^dm du

2^m13^j 2j−1

!

Γdm1Γd−1m−1 Γ2d−1m 2d−1

d−1 · 2^m13^j 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−1^2d−1 d^dd−1^d−1

m

≥

2d−1 2πd²m

2d−1^2d−1 d^dd−1^d−1

m

.

6.15

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

dm k0

−1^dmkgk dmk^2j

≤ d2d−12^m13^j√ 2πm

2j−1

!d−1d−1m−1√ 2d−1

d^dd−1^d−1 2d−1^2d−1

_m

. 6.16

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

−1^dmkgk dmk^2j

< 2^m23^j√ πd³ 2j−1

!d−1d−2√ m

d^dd−1^d−1 2d−1^2d−1

_m

. 6.18

Next, we need an upper bound for the Bernoulli numbersB2jcf.9, 23.1.15:

B2j≤ 2 2j

! 2π^2j

1−2^1−2j ≤ 4 2j

! 2π^2j,

j≥1

. 6.19

Letd≥3 andm≥max{2d−1, a/2}. Using this andLemma 6.1, we estimateEmin5.3:

|Em|< 3 2

√πd³ 2^m2 d−1d−2√

m

d^dd−1^d−1 2d−1^2d−1

_m

√πd³ 2^m2 d−1d−2√

m

d^dd−1^d−1 2d−1^2d−1

_m

×^dm

j1

B_2j 2j

1 a^2j − 1

b^2j 1 3^j

2j−1

! 3^2j 4j−1

!

≤

√πd³ 2^m2 d−1d−2√

m

d^dd−1^d−1 2d−1^2d−1

m

×

⎛

⎝3 2 ^dm

j1

4 2j−1

! 2π^2j

2·3^j 2j−1

! 3^2j 4j−1

!

⎞

⎠

< 4√ πd³ d−1d−2√

m

2d^dd−1^d−1 2d−1^2d−1

_m⎛

⎝3 2 8

∞ j1

⎛

⎝ √

3 2π

_2j

3 2π

_2j⎞

⎠

⎞

⎠

< 19√ πd³ d−1d−2√

m

2d^dd−1^d−1 2d−1^2d−1

_m .

6.20

Now, let

T_n:^an−1

j1

1 j −^bn−1

j1

1 j 2

n−1 j1

1 j −ⁿ

2−1

j1

1 j ⁿ

2−1

j1

e_j

j , n >1, 6.21

with the numberse_jintroduced in the proof ofTheorem 4.1. By definition ofR_nandS_nwe 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

−1^dmk

⎛

⎝2d−1mk−1 dm

⎞

⎠

⎛

⎝dm k

⎞

⎠T_kdm−γ−loga b

≤

dm k0

−1^dmkgkS_kdm−γ−loga b

dm k0

−1^dmkgkR_kdm

dm k0

−1^dmkgkS_kdm−γ−loga b |Em|

< c4·

1−1/d^d d−14^d

_m

19√ πd³ d−1d−2√

m

2d^dd−1^d−1 2d−1^2d−1

_m .

6.22

By similar arguments we get the same bound whenb > a. Ford≥3 it can easily be seen that

2d^dd−1^d−1

2d−1^2d−1 22d−1

d−1 · 1−1/d^d 1−1/2d^2d · 1

4^d < 18

4^d1. 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 −ⁿ

2−1

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

−1^k

2d−1mk−1 dm

dm k

T_kdm−γ−loga b < √c5

m 18

4^d1 _m

, m≥1.

6.25

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

−1^dmk

2d−1mk−1 dm

dm k

T_kdm−γ−loga b

< c6·

1−1/d^d d−14^d

_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

−1^dmkgk dmk²

₁

0

₁

0

1−u^dm1−w^mu^2d−1m−1w^d−1m−1 1−1−uw^dm1 du dw

₁

0

₁

0

1−u²1−wu² 1−1−uw³

1−uu²w 1−1−uw

_d−2

×

1−u^d1−wu^2d−1w^d−1 1−1−uw^d

_m−1 du dw

≤ 1 4^d−2

1−1/d^d d−14^d

_m−11

0

₁

0

1−u²1−wu² 1−1−uw³ du dw 2d−1

31−1/d^d ·

1−1/d^d d−14^d

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−u^d1−wu^2d−1w^d−1

1−1−uw^d , 7.3

take their maximum values 4^2−dand1−1/d^d/d−14^dinside 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/d^d d−14^d

m

, d≥42, m≥1, 7.4