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 constant*γ*. Here,
we discuss in detail recently published results of A. I. Aptekarev and T. Rivoal who found
rational approximations to*γ*and*γ*log*qq*∈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 approximate*γ*and*γ*log*q*.
It is shown that for every*q*∈ Q>0and every integer*d* ≥ 42 there are infinitely many rationals

*am/bm*for*m*1*,*2*, . . .*such that|*γ*log*q*−*am/bm*| 1−1*/dd/d*−14*d*
*m*

and*bm*|*Zm*with
log*Zm*∼12*d*2*m*2for*m*tending 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

−log*n* *n*≥2*.* 1.1

It is well known that the sequence*snn*≥1converges to Euler’s constant*γ*0*,*577*. . .*, where

*snγ*O

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 series*snn*≥1 with

integer coeﬃcients which improves the rate of convergence. Let*τ* be an additional positive

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

*n*

*k*0
−1*nk*

*nkτ*−1
*n*

*n*

*k*

·*skτ*−*γ*

≤

*τ*−1!

2*nn*1*n*2· · ·*nτ.* 1.3

*Particularly, by choosingτn*≥*2, one gets the following result.*

* Corollary 1.2. For any integern*≥

*2one has*

*n*

*k*0
−1*nk*

2*nk*−1
*n*

*n*

*k*

·*snk*−*γ*
≤

1

2*n*22*n*

*n*

≤ 1

*n*3*/*2_{·}_{4}*n.* 1.4

Some authors have generalized the result ofProposition 1.1under various aspects. At

first one cites a result due to Rivoal2.

**Proposition 1.3**see2*. Forntending to infinity, one has*

*γ*−

1

−2*n*

*n*

*k*0
−1*k*

2*n*2*k*
*n*

*n*

*k*

*s*2*kn*1

O

1
*n*27*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 1.4**3*. Forntending to infinity the following asymptotic formula holds:*

*γ*−

*n*

*k*0
−1*nk*

*nk*
*n*

*n*

*k*

*skn*1

4*n*1*on.* 1.6

Recently the author has found series transformations involving three parameters*n*,*τ*1

and*τ*2,4. In Propositions1.5and1.6certain integral representations of thediscreteseries

transformations are given, which exhibit importantanalytical tools to estimate the error

terms of the transformations.

**Proposition 1.5**see4*. Letn*≥*1,τ*1≥1*,andτ*2 ≥*1 be integers. Additionally one assumes that*

*Then one has*

*n*

*k*0
−1*nk*

*nτ*1*k*

*n*

*n*

*k*

·*skτ*2−*γ*

−1*n*1

_{1}

0

1

1−*u*

1
*log u*

·*uτ*2−*τ*1−1· *∂*

*n*

*∂un*

*unτ*1_{1}−_{u}n*n*!

*du.*

1.8

**Proposition 1.6**see4*. Letn*≥*1,τ*1≥*1 andτ*2≥*1 be integers. Additionally one assumes that*

1*τ*1 ≤ *τ*2 ≤ 1*nτ*1*.* 1.9

*Then one has*

*n*

*k*0
−1*nk*

*nτ*1*k*

*n*

*n*

*k*

·*skτ*2−*γ*

−1*nτ*2−*τ*1

_{1}

0

_{1}

0

*wt*·1−*u*

*nτ*1_{u}n_{1}−* _{t}τ*2−

*τ*1−1

*1−*

_{t}nτ*τ*21

1−*utn*1 *du dt,*

1.10

*with*

*wt*: 1

*t*·

*π*2_{}_{log}2

1
*t* −1

*.* 1.11

Setting

*nτ*2*dm,* *τ*1 *d*−1*m*−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*

*k*0

−1*dmk*

2*d*−1*mk*−1
*dm*

*dm*

*k*

·*skdm*−*γ*
*< Cd*·

1−1*/dd*

*d*−14*d*
m−2

*,* 1.13

*where 0< Cd*≤1*/*16*π*2*is some constant depending only ond. Ford2 one gets*

2*m*

*k*0
−1*k*

3*mk*−1

2*m*

2*m*

*k*

·*sk*2*m*−*γ*
*<*

16
7*π*

2

· 1

*For an application ofCorollary 1.7let the integersBmandAmbe defined by*

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

*Am*:*Bm*

2*m*

*k*0
−1*k*

3*mk*−1

2*m*

2*m*

*k*

·

1 1

2· · ·
1
*k*2*m*−1

*.*

1.15

Λ*kdenotes the von Mangoldt function. By [5, Theorem 434] one has*

*ψm*:

*k*≤*m*

Λ*k*∼*m.* 1.16

*Then, forε*: log 55*/*4−1*>*0*.0018, there is some integerm*0*such that*

*Bmeψ*4*m* *< e*41*εm*55*m* *m*≥*m*0*.* 1.17

*Multiplying*1.14*byBm, we deduce the following corollary.*

* Corollary 1.8. There is an integerm*0

*such that one has for all integersm*≥

*m*0

*that*

*Bm*

2*m*

*k*0
−1*k*

3*mk*−1

2*m*

2*m*

*k*

·log*k*2*m* *γBm*−*Am*
*<*

16
7*π*

2

·

55 64

m

*.* 1.18

**2. Results on Rational Approximations to**

**γ**

**γ**

In 2007, Aptekarev and his collaborators6found rational approximations to*γ*, which are

based on a linear third-order recurrence. For the sake of brevity, let*Dn* l.c.m.1*,*2*, . . . , n*.

**Proposition 2.1**see6*. Letpnn*≥0*andqnn*≥0*be two solutions of the linear recurrence*

16*n*−15*n*1*un*1

128*n*340*n*2−82*n*−45 *un*

−*n*256*n*3−240*n*264*n*−7 *un*−1*nn*−116*n*1*un*−2

2.1

*withp*0*0,p*1*2,p*231*/2 andq*0*1,q*1*3,q*2*25. Then, one hasqn*∈Z*,Dnpn*∈Z*, and*

*γ*−*pn*

*qn*
∼*c*0*e*−2

√

2*n _{,}*

_{q}*n*∼

*c*1

*n*1*/*4
2*n*!

*n*! *e*

√

2*n _{,}*

_{}

_{2.2}

_{}

It seems interesting to replace the fraction*pn/qn*by

*An*

*Bn* :

*Dnpn*

*Dnqn,* 2.3

and to estimate the remainder in terms of*Bn*.

* Corollary 2.2. Let 0< ε <1. Then there are two positive constantsc*2

*, c*3, such that for all su

*ﬃciently*

*large integersnone has*

*c*2exp

−21*ε*√2

log*Bn/*log log*Bn*

*<γ*−*An*
*Bn*

*< c*3exp

−21−*ε*√2

log*Bn/*log log*Bn*

*.*

2.4

Recently, Rivoal 7 presented a related approach to the theory of rational

approxi-mations to Euler’s constant*γ*, 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 inQ*z*that converges subexponentially to log*z* *γ*for any complex

number*z*∈C\−∞*,*0. Here, log is defined by its principal branch. We cite a corollary from

7.

**Proposition 2.3**see7*. (i) The recurrence*

*n*328*n*118*n*19*Un*3

24*n*2_{}_{145}_{n}_{}_{215} _{}_{8}_{n}_{}_{11}_{}_{U}

*n*2

−24*n*3105*n*2124*n*25 8*n*27*Un*1

*n*228*n*198*n*27*Un,*

2.5

*provides two sequences of rational numberspnn*≥0*andqnn*≥0*withp*0−*1,p*1*4,p*277*/4 and*

*q*0*1,q*1 *7,q*265*/2 such thatpn/qnn*≥0*converges toγ.*
*(ii) The recurrence*

*n*1*n*2*n*3*U _{n}*

_{}

_{3}

3*n*219*n*29 *n*1*Un*2

−3*n*3_{}_{6}* _{n}*2

_{−}

_{7}

_{n}_{−}

_{13}

_{U}*n*1 *n*23*Un,*

2.6

*provides two sequences of rational numberspnn*≥0*andqnn*≥0*withp*0 −*1,p*1 *11,p*2*71 and*

The goal of this paper is to construct rational approximations to*γ*log*a/b*without
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 by*B*2*n*the Bernoulli numbers, that is,*B*21*/*6,*B*4−1*/*30,

*B*6 1*/*42, and so onIn Sections3–6the Bernoulli numbers cannot be confused with the

integers*Bn*fromCorollary 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

*j*1

1
*j* −

*bn*−1

*j*1

1
*j* 2

*n*−1

*j*1

1
*j* −

*n*_{}2_{−}_{1}

*j*1

1

*j* −

1
2*n*2

*dm*

*j*1

*B*2*j*

2*j*

1
*n*2*j*

1
*a*2*j* −

1
*b*2*j* 1

− 1

*n*4*j*

*,* *n*≥1*.*

2.7

*Then,*

*dm*

*k*0

−1*dmk*

2*d*−1*mk*−1
*dm*

*dm*

*k*

*Skdm*−*γ*−log*a*

*b*

*< c*4·

1−1*/dd*

*d*−14*d*
*m*

*,* 2.8

*wherec*4*is some positive constant depending only ond.*

**3. Proof of**

**Theorem 2.4**

**Lemma 3.1. One has for positive integers**dandm

*gk*:

2*d*−1*mk*−1
*dm*

*dm*

*k*

*<*16*dm* 0≤*k*≤*dm.* 3.1

*Proof. Applying the well known inequalityg*

*h* ≤2

*g*_{, we get}

2*d*−1*mk*−1
*dm*

*dm*

*k*

≤22*d*−1*mdm*−1_{2}*dm*_{}_{2}4*dm*−*m*−1_{<}_{16}*dm _{.}*

_{}

_{3.2}

_{}

This proves the lemma.

*gk*takes its maximum value for*kk*0with

*k*0

√

5*d*2_{−}_{4}_{d}_{}_{1}_{−}_{d}_{}_{1}

which leads to a better bound than 16*dm*inLemma 3.1. But we are satisfied withLemma 3.1.

A main tool in provingTheorem 2.4is Euler’s summation formula in the form

*n*

*i*1

*fi*

_{n}

1

*fxdxf*1 *fn*

2

*r*

*j*1

*B*2*j*

2*j*!

*f*2*j*−1*n*−*f*2*j*−11 *Rr,* 3.4

where*r* ∈ Nis a suitable chosen parameter, and the remainder*Rr* is defined by a periodic

Bernoulli polynomial*P*2*r*1*x*, namely

*Rr* _{} 1

2*r*1!

*n*

1

*P*2*r*1*xf*2*r*1*xdx,* 3.5

with

*P*2*r*1*x* −1*r*−12*r*1!

∞

*j*1

2 sin2*πjx*

2*πj*2*r*1 *.* 3.6

Applying the summation formula to the function*fx* 1*/x*, we getsee8, equation5

*n*−1

*i*1

1

*i* log*n*

1

2−

1

2*n*

*r*

*j*1

*B*2*j*

2*j*

1− 1

*n*2*j*

−
_{n}

1

*P*2*r*1*x*

*x*2*r*2 *dx,* *n, r* ∈N*.* 3.7

It follows that

*n*_{}2_{−}_{1}

*in*

1

*i* −log*n*
1
2*n*−

1
2*n*2

*r*

*j*1

*B*2*j*

2*j*

1
*n*2*j* −

1
*n*4*j*

−

*n*2

*n*

*P*2*r*1*x*

*x*2*r*2 *dx,* *n, r* ∈N*.* 3.8

We proveTheorem 2.4for*a*≥*b*. The case*a < b*is 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

2*n*

*r*

*j*1

*B*2*j*

2*jn*2*j*

1
*b*2*j* −

1
*a*2*j*

−
_{an}

*bn*

*P*2*r*1*x*

*x*2*r*2 *dx,* *n, r*∈N*.*

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

* _{n}*2

*n*

*P*2*r*1*x*

*x*2*r*2 *dx*

≤

* _{n}*2

*n*

|*P*2*r*1*x*|

*x*2*r*2 *dx*≤

_{∞}

*n*

|*P*2*r*1*x*|

*x*2*r*2 *dx*

≤ 22*r*1!

_{∞}

*n*

1
*x*2*r*2

∞

*j*1

1

2*πj*2*r*1*dx*

22*r*1!
2*π*2*r*1

−_{} 1

2*r*1*x*2*r*1

∞

*xn*

∞

*j*1

1
*j*2*r*1

22*r*!

2*π*2*r*1*n*2*r*1 *ζ*2*r*1*<*

32*r*!

2*π*2*r*1*n*2*r*1*,*

3.10

since 2*ζ*2*r*1 ≤ 2*ζ*3 *<* 3. Next, we assume that*n* ≥ *a*. Hence *bn, an* ⊆ *n, n*2_{}_{, and}

therefore we estimate the integral on the right-hand side in3.9by

*an*

*bn*

*P*2*r*1*x*

*x*2*r*2 *dx*

≤

_{an}

*bn*

|*P*2*r*1*x*|

*x*2*r*2 *dx*

≤
* _{n}*2

*n*

|*P*2*r*1*x*|

*x*2*r*2 *dx*≤

32*r*!

2*π*2*r*1*n*2*r*1*.*

3.11

In the sequel we put*r* *dm*. Moreover, in the above formula we now replace*n*by*dmk*

with 0≤*k*≤*dm*. In order to estimate2*r*! we use Stirling’s formula

√

2*πm*

*m*
*e*

*m*

*< m*!*<*

2*πm*1

*m*
*e*

*m*

*,* *m >*0*.* 3.12

Then, it follows that

_{}_{dm}_{}_{k}_{}2

*dmk*

*P*2*r*1*x*

*x*2*r*2 *dx*

≤

32*r*!

2*π*2*r*1*dmk*2*r*1 ≤

32*r*!

2*π*2*r*1*dm*2*r*1

32*dm*!
2*π*2*dm*1*dm*2*dm*1

≤ 3

*π*2*dm*1

2*π dm*2*dm*1 ·

2*dm*

*e*

2*dm*

≤ 3

√

3*πdm*

2*πdmπe*2*dm,*

and similarly we have

*admk*

*bdmk*

*P*2*r*1*x*

*x*2*r*2 *dx*

≤

3√3*πdm*

2*πdmπe*2*dm,* *dm*≥*a.* 3.14

By using the definition of*Sn*inTheorem 2.4, the formula1.1for*sn*, and the identities3.8,

3.9, it follows that

*Sn*−*γ*−log*a*

*b*

*sn*−*γ*

*sn*−*s*2*n* *san*−*sbn*− 1

2*n*2

*r*

*j*1

*B*2*j*

2*j*

1
*n*2*j*

1
*a*2*j* −

1
*b*2*j* 1

− 1

*n*4*j*

*sn*−*γ*
1
2*n*
1
*b*−
1
*a*−1

* _{n}*2

*n*

*P*2*r*1*x*

*x*2*r*2 *dx*−

_{an}

*bn*

*P*2*r*1*x*

*x*2*r*2 *dx,*

3.15

where*r* is specified to*r* *dm*and*n*to*n* *dmk*. Moreover, we know from4, Lemma 2

that

*dm*

*k*0

−1*dmkgk* 1*,* *m*≥1*.* 3.16

By setting*ndmk*, the above formula for the series transformation of*Sdmk*simplifies to

*dm*

*k*0

−1*dmkgkSdmk*−*γ*−log

*a*
*b*
_{}*dm*

*k*0

−1*dmkgksdmk*−*γ*
1
2
1
*b*−
1
*a*−1

*dm*

*k*0

−1*dmkgk*

*dmk*

*dm*

*k*0

−1*dmkgk*

_{}_{dm}_{}_{k}_{}2

*dmk*

*P*2*r*1*x*

*x*2*r*2 *dx*

−*dm*
*k*0

−1*dmkgk*

*admk*

*bdmk*

*P*2*r*1*x*

*x*2*r*2 *dx*

≤_{}*dm*
*k*0

−1*dmkgksdmk*−*γ*
_{}

*dm*

*k*0

*gk*

_{}*dmk*2

*dmk*

*P*2*r*1*x*

*x*2*r*2 *dx*

*dm*

*k*0

*gk*_{}

*admk*

*bdmk*

*P*2*r*1*x*

*x*2*r*2 *dx*

*< Cd*·

1−1*/dd*

*d*−14*d*

*m*−2
*dm*

*k*0

*gk* 3

√

3*πdm*

*πdmπe*2*dm,*

where*dm* ≥*a, m*≥2, and*d* ≥3. Here, we have used the results fromCorollary 1.7,3.13,

and3.14. The sum

*dm*

*k*0

−1*dmkgk*

*dmk* 3.18

vanishes, since for every real number*x >*−*dm*we have

*dm*

*k*0

−1*dmk*2*d*−1*mk*−1

*dm*

*dm*
*k*

*dmkx*

1−*d*−1*mx*· · ·*mx*

*dmx _{dm}*

_{}

_{1}

*,*3.19

where on the right-hand side for an integer*x*with−*m* ≤ *x* ≤ *d*−1*m*−1 one term in the

numerator equals to zero. The inequality

64

*πe*2

d

*<* 1−1*/d*

*d*

2*d*−1 3.20

holds for all integers*d*≥42. Now, usingLemma 3.1, we estimate the right-hand side in3.17

for*dm*≥*a*and*d*≥42 as follows:

*dm*

*k*0

−1*dmkgkSdmk*−*γ*−log

*a*
*b*

*< Cd*·

1−1*/dd*

*d*−14*d*

*m*−2
*dm*

*k*0

3√3*πdm*

*πdm*

16*dm*

*πe*2*dm*

*Cd*·

1−1*/dd*

*d*−14*d*

*m*−2

3*dm*1

√

3*dm*

*dm*√*π*

1
4*dm*

64

*πe*2

*dm*

3*.*20

*< Cd*·

1−1*/dd*

*d*−14*d*

m−2

3*dm*1

√

3*dm*

*dm*2*m*√_{π}

1−1*/dd*

*d*−14*d*
m

≤*Cd*·

1−1*/dd*

*d*−14*d*

m−2 85

28

3*d*
*π*

1−1*/dd*

*d*−14*d*
m

≤*c*4

1−1*/dd*

*d*−14*d*
m

*.*

3.21

The last but one estimate holds for all integers*m*≥2,*d*≥42, and*c*4is a suitable positive real

**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*

*k*0

−1*dmkgkSkdm,* 4.1

for*m*tending to infinity, where*am*∈Zand*bm*∈Nare coprime integers.

* Theorem 4.1. For everym*≥

*1 there is an integerZmwithZm>0,bm*|

*Zm, and*

log*Zm*∼12*d*2*m*2*,* *m*−→ ∞*.* 4.2

*Proof. We will need some basic facts on the arithmetical functionsϑx*and*ψx*. Let

*ϑx*

*p*≤*x*

log*p,* *x >*1*,*

*ψx*

*p*≤*x*

log*x*

log*p*

log*p,* *x >*1*,*

4.3

where*p*is restricted on primes. Moreover, let*Dn*: l.c.m1*,*2*, . . . , n*for positive integers*n*.

Then,

*ψn* log*Dn,* *n*≥1*,* 4.4

*ψx*∼ *ϑx*∼ *πx*log*x* ∼ *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*2*k*: The denominators of *B*2*k* are squarefree, and they are divisible

exactly by those primes*p*with*p*−1|2*k*. Hence,

*B*2*k*

*p*≤2*k*1

*p* ∈ Z*,* *k*1*,*2*, . . ..* 4.6

Next, let max{*a, b*} ≤*dm*≤*n*≤2*dmnkdm*are the subscripts of*Skdm*inTheorem 2.4.

First, we consider the following terms from the series transformation in*Sm*:

*an*−1

*j*1

1
*j* −

*bn*−1

*j*1

1
*j* 2

*n*−1

*j*1

1
*j* −

*n*_{}2_{−}_{1}

*j*1

1
*j* :

*n*_{}2_{−}_{1}

*j*1

*ej*

with

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

1*,* if 1≤*j*≤*n*−1

−1*,* if*n*≤*j*≤*bn*−1

0*,* if *bn*≤*j*≤*an*−1

−1*,* if*an*≤*j*≤*n*2_{−}_{1}

*a*≥*b*,

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

1*,* if 1≤*j*≤*n*−1

−1*,* if*n*≤*j*≤*an*−1

−2*,* if*an*≤*j*≤*bn*−1

−1*,* if*bn*≤*j* ≤*n*2_{−}_{1}_{.}

*a < b*.

4.8

For every*m*≥1 there is a rational*xm/ym*defined by

*xm*

*ym*

*dm*

*k*0

−1*dmkgk*

*kdm*2_{−}_{1}

*j*1

*ej*

*j* *,* 4.9

where*xm*∈Z,*ym*∈N,*xm, ym* 1, and

*ym*|*Ym* :*D*4*d*2* _{m}*2

*,*

*dm*≥max{

*a, b*}

*.*4.10

Similarly, we define rationals*um/vm*by

*um*

*vm*
*dm*

*k*0

−1*dmkgk*

× ⎛

⎝_{−} 1

2*kdm*2

*dm*

*j*1

*B*2*j*

2*j*

1

*kdm*2*j*

1
*a*2*j* −

1
*b*2*j* 1

− 1

*kdm*4*j*

⎞
⎠_{,}

4.11

where*um*∈Z,*vm*∈Nand*um, vm* 1. We have

*kdm*2*j*|*kdm*4*dm,* 0≤*k*≤*dm,* 1≤*j* ≤*dm.* 4.12

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

*vm*|*Vm* :2*ab*2*dmDdm*
⎛

⎝

*p*≤2*dm*1

*p*

⎞

⎠_{}*D*2*dm*4*dm,* *dm*≥max{*a, b*}*.* 4.13

Note that*D*2*dm* l.c.m.*dm, . . . ,*2*dm*, since every integer*n*1with 1 ≤ *n*1 *< dm*divides at

From4.10and4.13we conclude on

*bm*|*Zm*:2*ab*2*dmDdmD*4*d*2* _{m}*2

*D*

_{2}

*4*

_{dm}*dm*

⎛

⎝

*p*≤2*dm*1

*p*

⎞

⎠*.* 4.14

Hence we have from4.4and4.5that

log*Zm*log 22*dm*log*ab* *ψdm* *ψ*

4*d*2*m*2 4*dmψ*2*dm* *ϑ*2*dm*1

∼ log 22*dm*log*ab* *dm*4*d*2* _{m}*2

_{}

_{8}

*2*

_{d}*2*

_{m}_{2}

_{dm}_{}

_{1}

_{}

1log 232 log*abdm*12*d*2*m*2

∼12*d*2*m*2 *m*−→ ∞*.*

4.15

The theorem is proved.

*Remark 4.2. On the one side we have shown that logYm*∼4*d*2*m*2and log*Vm*∼8*d*2*m*2. On the

other side, every prime*p*dividing*Vm*satisfies*p*≤max{*a, b, dm,*2*dm*1*,*2*dm*}2*dm*1 and

therefore*p*divides*Ym* *D*4*d*2* _{m}*2. Conversely, all primes

*p*with 2

*dm*1

*< p <*4

*d*2

*m*2divide

*Ym*, but not*Vm*. That means:*Vm*is much bigger than*Ym*, but*Vm* is formed by powers of small

primes, whereas*Ym* is divisible by many big primes.

**5. Simplification of the Transformed Series**

Let

*Rn*:−

1
2*n*2

*dm*

*j*1

*B*2*j*

2*j*

1
*n*2*j*

1
*a*2*j* −

1
*b*2*j* 1

− 1

*n*4*j*

*,* 5.1

such that

*Sn*
*an*−1

*j*1

1
*j* −

*bn*−1

*j*1

1
*j* 2

*n*−1

*j*1

1
*j* −

*n*_{}2_{−}_{1}

*j*1

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 of*Rkdm*to the series transformation inTheorem 2.4. For this purpose, we define

*Em*:
*dm*

*k*0

−1*dmk*

2*d*−1*mk*−1
*dm*

*dm*

*k*

*Rkdm* −1

2

*dm*

*k*0

−1*dmkgk*

*dmk*2

*dm*

*j*1

*B*2*j*

2*j*

1
*a*2*j* −

1
*b*2*j* 1

*dm*

*k*0

−1*dmkgk*

*dmk*2*j* −

*dm*

*k*0

−1*dmkgk*

*dmk*4*j*

*.*

5.3

**Lemma 5.1. For positive integers**d, jandmone has

*dm*

*k*0

−1*dmkgk*

*dmk*2*j* −

−1*dm*

*dm*!2*j*−1!

1

0

*um*log*u*2*j*−1 *∂*

*dm*

*∂udm*

*u*2*d*−1*m*−11−*udm* *du.* 5.4

*Proof. For integersk, r*and a real number*ρ*with*kρ >*0 the identity

1

*kρr*
1

*r*−1!

_{∞}

0

*e*−*kρttr*−1 *dt* 5.5

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

Introducing the new variable*u*:*e*−*t*_{, we then get}

2*m*

*k*0

−1*dmkgk*

*kdm*2*j* −

−1*dm*

2*j*−1!

*dm*

*k*0

−1*kgk*

1

0

*ukdm*−1log*u*2*j*−1 *du*

−−1*dm*

2*j*−1!

_{1}

0

_{}_{dm}

*k*0

−1*kgkukd*−1*m*−1

*um*log*u*2*j*−1 *du.*

5.6

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

*n*

*k*0
−1*k*

*nτk*

*n*

*n*

*k*

*uτk* *∂*

*n*

*∂un*

*unτ*_{}_{1}_{−}_{u}_{}*n*

*n*!

*,* *n, τ*∈N∪ {0}*,* 5.7

in which we put*ndm*and*τ* *d*−1*m*−1. This gives the identity stated in the lemma.

The following result deals with the case*j*1, in which we express the finite sum by a double

integral on a rational function.

**Corollary 5.2. For every positive integer**mone has

*dm*

*k*0

−1*dmkgk*

*dmk*2 −1

*d*−1*m*
1

0

1

0

1−*udm*1−*wmu*2*d*−1*m*−1* _{w}d*−1

*m*−1

1−1−*uwdm*1 *du dw.* 5.8

*Proof. Setj* 1 inLemma 5.1, and note that

log*u*−1−*u*

_{1}

0

*dw*

Hence,

*dm*

*k*0

−1*dmkgk*

*dmk*2 −

−1*dm*

*dm*!

1

0

*um*log*u* *∂*

*dm*

*∂udm*

*u*2*d*−1*m*−11−*udm* *du*

_{}−1*dm*

*dm*!

1

0

1

0

1−*uum*

1−1−*uw*
*∂dm*

*∂udm*

*u*2*d*−1*m*−11−*udm* *du dw.*

5.10

Let *s* be any positive integer. Then we have the following decomposition of a rational

function, in which*u*is considered as variable and*w*as parameter:

*us*

1−1−*uw*

*s*−1

*ν*0

*w*−1*ν*
*wν*1 *u*

*s*−*ν*−1_{}

*w*−1
*w*

*s* _{1}

1−1−*uw.* 5.11

We additionally assume that*s*−1 *< dm*. Then, diﬀerentiating this identity*dm*-times with

respect to*u*, the polynomial in*u*on the right-hand side vanishes identically:

*∂dm*

*∂udm*

*us*

1−1−*uw*

*w*−1
*w*

*s* _{}_{−}_{1}_{}*dm*_{}_{dm}_{}_{!} _{w}dm

1−1−*uwdm*1*.* 5.12

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

*dm*

*k*0

−1*dmkgk*

*dmk*2
1

*dm*!

1

0

1

0

*u*2*d*−1*m*−11−*udm* *∂*

*dm*

*∂udm*

*um*−*um*1
1−1−*uw*

*du dw*

_{} 1

*dm*!

_{1}

0

_{1}

0

*u*2*d*−1*m*−11−*udm*

_{w}_{−}_{1}

*w*

*m*
−

*w*−1
*w*

*m*1

× −1*dmdm*!*wdm* *du dw*

1−1−*uwdm*1 *.*

5.13

The corollary is proved by noting that

*w*−1
*w*

*m*
−

*w*−1
*w*

*m*1

−1*m* 1−*w*

*m*

**6. Estimating**

**E**

**m**

In this section we estimate*Em* defined in5.3. Substituting 1−*u*for*u*into the integral in

Lemma 5.1and applying iterated integration by parts, we get

*dm*

*k*0

−1*dmkgk*

*dmk*2*j*

− −1*dm*

*dm*!2*j*−1!

_{1}

0

*∂dm*

*∂udm*

1−*um*log1−*u*2*j*−1 1−*u*2*d*−1*m*−1*udm* *du.*

6.1

Set

*fu*: 1−*um*log1−*u*2*j*−1*,* 6.2

where*m*and*j*are kept fixed. We have*f*0 0. For an integer*k >*0 we use Cauchy’s formula

*fka* *k*!
2*πi*

*C*

*fz*

*z*−*ak*1 *dz* 6.3

to estimate |*fk*_{}_{0}_{}_{|}_{. Let}_{C}_{denote the circle in the complex plane centered around 0 with}

radius*R*:1−1*/*2*k*. With*a*0 and*fz*defined above, Cauchy’s formula yields the identity

*fk*0 *k*!
2*πRk*

*π*

−*π*

*e*−*ikφ*1−*Reiφ* mlog2*j*−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 arguments*φ* of a complex

number*z /*∈−∞*,*0are taken from the interval−*π, π*. Therefore, using 1−*Reiφ*1−*R*cos*φ*−
*iR*sin*φ*, we get

1−*Reiφ* 1*R*2_{−}_{2}_{R}_{cos}_{φ}_{}_{:}_{A}_{R, φ}_{,}

arg1−*Reiφ* −arctan

_{R}_{sin}_{φ}

1−*R*cos*φ*

*.*

6.5

Hence,

log1−*Reiφ* _{}_{ln}_{1}_{}* _{R}*2

_{−}

_{2}

_{R}_{cos}

_{φ}_{−}

_{i}_{arctan}

*R*sin*φ*

1−*R*cos*φ*

1

2ln

*AR, φ*−*i*arctan

_{R}_{sin}_{φ}

1−*R*cos*φ*

*.*

Thus, it follows from6.4that

*fk*0≤ *k*!

2*πRk*
_{π}

−*π*

1−*Reiφm*·log1−*Reiφ *2*j*−1 *dφ*

*k*!

2*πRk*
_{π}

−*π*

*AR, φ*m/2

1 4ln

2_{A}_{}_{R, φ}_{}_{}_{arctan}2

_{R}_{sin}_{φ}

1−*R*cos*φ*

2*j*−1*/*2

*dφ*

*k*!

*πRk*
*π*

0

*AR, φm/*2

1 4ln

2_{A}_{}_{R, φ}_{}_{}_{arctan}2

*R*sin*φ*

1−*R*cos*φ*

2*j*−1*/*2

*dφ.*

6.7

From 0*< R <*1 we conclude on

0*<*1−*R*21*R*2−2*R*≤1*R*2−2*R*cos*φAR, φ<*4*,* 0≤*φ*≤*π,*

0≤ *R*sin*φ*

1−*R*cos*φ* ≤

sin*φ*
1−cos*φ,*

0*< φ*≤*π.*

6.8

Since arctan is a strictly increasing function, we get

arctan

_{R}_{sin}_{φ}

1−*R*cos*φ*

≤arctan

_{sin}_{φ}

1−cos*φ*

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 for*φ*0. Finally, we note that*Rk* _{1}_{−}_{1}_{/}_{2}_{k}_{}*k*_{≥}

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

*fk*0≤ *k*!

*πRk*
_{π}

0

4*m/*2

ln24

4 arctan

2

_{sin}_{φ}

1−cos*φ*

2*j*−1*/*2

*dφ*

≤ 2*m*1*k*!

*π*

_{π}

0

ln22

_{π}_{−}_{φ}

2

2 2*j*−1*/*2

*dφ*

≤ 2*m*1*k*!

*π*

*π*

0

ln22*π*

2

4

*j*−1*/*2

*dφ*

≤ 2*m*1*k*!

*π*

_{π}

0

3*j*−1*/*2 _{dφ}_{≤} _{2}*m*1_{3}*j _{k}*

_{!}

_{.}It follows that the Taylor series expansion of*fu*,

*fu*

∞

*k*0

*fk*_{}_{0}_{}

*k*! *u*

*k _{,}*

_{}

_{6.11}

_{}

converges at least for−1*< u <*1. Then,

*fdmu*

∞

*kdm*

*fk*_{}_{0}_{}

*k*−*dm*! *u*

*k*−*dm*_{}∞
*k*0

*fkdm*_{}_{0}_{}

*k*! *u*

*k _{,}*

_{}

_{6.12}

_{}

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

*fdmu*≤

∞

*k*0

*fkdm*_{}_{0}_{}

*k*! *u*

*k* _{≤} _{2}*m*1_{3}*j* ∞
*k*0

*kdm*!

*k*! *u*

*k*

2*m*1_{3}*j*_{}_{dm}_{}_{!}∞
*k*0

*kdm*

*k*

*uk*_{} 2*m*13*jdm*!

1−*udm*1 *.*

6.13

Combining6.13with the result from6.1, we get for*m >*1

*dm*

*k*0

−1*dmkgk*

*dmk*2*j*

≤

2*m*1_{3}*j*

2*j*−1!

_{1}

0

1−*ud*−1*m*−2*udm* *du*

2*m*13*j*

2*j*−1!

Γ*dm*1Γ*d*−1*m*−1

Γ2*d*−1*m*

2*d*−1

*d*−1 ·

2*m*1_{3}*j*

2*j*−1!*d*−1*m*−1 ·
1

_{}_{2}_{d}_{−}_{1}_{}_{m}

*dm*

*.*

6.14

We estimate the binomial coeﬃcient by Stirling’s formula 3.12. For this purpose we

additionally assume that*m*≥2*d*−1:

2*d*−1*m*
*dm*

≥

2*d*−1*m*

2*πdm*1*d*−1*m*1

2*d*−12*d*−1
*dd*_{}_{d}_{−}_{1}_{}*d*−1

m

≥

2*d*−1
2*πd*2_{m}

2*d*−12*d*−1
*dd _{d}*−

_{1}

*d*−1

m

*.*

6.15

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

*dm*

*k*0

−1*dmkgk*

*dmk*2*j*

≤

*d*2*d*−12*m*1_{3}*j*√_{2}_{πm}

2*j*−1!*d*−1*d*−1*m*−1√2*d*−1

*dd*_{}_{d}_{−}_{1}_{}*d*−1
2*d*−12*d*−1

*m*

For all integers*m*≥1 and*d*≥1 we have

2*d*−1√2*πm*

√

2*d*−1 *<*2

*πdm,* *d*−1*d*−1*m*−1≥*d*−1*d*−2*m.* 6.17

Thus we have proven the following result.

* Lemma 6.1. For all integersd, mwithd*≥

*3 andm*≥2

*d*−

*1 one has*

*dm*

*k*0

−1*dmkgk*

*dmk*2*j*

*<*

2*m*2_{3}*j*√* _{πd}*3

2*j*−1!*d*−1*d*−2√*m*

*dd*_{}_{d}_{−}_{1}_{}*d*−1
2*d*−12*d*−1

*m*

*.* 6.18

Next, we need an upper bound for the Bernoulli numbers*B*2*j*cf.9, 23.1.15:

_{B}_{2}_{j}_{≤} 22*j*!

2*π*2*j*1−21−2*j* ≤

42*j*!

2*π*2*j,*

*j*≥1*.* 6.19

Let*d*≥3 and*m*≥max{2*d*−1*, a/*2}. Using this andLemma 6.1, we estimate*Em*in5.3:

|*Em*|*<*

3 2

√

*πd*3 _{2}*m*2
*d*−1*d*−2√*m*

*dd*_{}_{d}_{−}_{1}_{}*d*−1
2*d*−12*d*−1

*m*

√

*πd*3 _{2}*m*2
*d*−1*d*−2√*m*

*dd*_{}_{d}_{−}_{1}_{}*d*−1
2*d*−12*d*−1

*m*

×*dm*
*j*1

*B*2*j*

2*j*

* _{a}*12

*j*−

1
*b*2*j* 1

3*j*

2*j*−1!
32*j*

4*j*−1!

≤ √

*πd*3 _{2}*m*2
*d*−1*d*−2√*m*

*dd*_{}_{d}_{−}_{1}_{}*d*−1
2*d*−12*d*−1

m

× ⎛ ⎝3

2

*dm*

*j*1

42*j*−1!

2*π*2*j*

2·3*j*

2*j*−1!
32*j*

4*j*−1!

⎞ ⎠

*<* 4

√

*πd*3
*d*−1*d*−2√*m*

2*dd*_{}_{d}_{−}_{1}_{}*d*−1
2*d*−12*d*−1

*m*⎛
⎝3

2 8

∞

*j*1

⎛ ⎝

√

3
2*π*

2*j*

3
2*π*

2*j*⎞
⎠

⎞ ⎠

*<* 19

√

*πd*3
*d*−1*d*−2√*m*

2*dd*_{}_{d}_{−}_{1}_{}*d*−1
2*d*−12*d*−1

*m*

*.*

Now, let

*Tn*:
*an*−1

*j*1

1
*j* −

*bn*−1

*j*1

1
*j* 2

*n*−1

*j*1

1
*j* −

*n*_{}2_{−}_{1}

*j*1

1

*j*

*n*_{}2_{−}_{1}

*j*1

*ej*

*j* *,* *n >*1*,* 6.21

with the numbers*ej*introduced in the proof ofTheorem 4.1. By definition of*Rn*and*Sn*we

then have*Tn* *Sn*−*Rn*, and therefore we can estimate the series transformation of *Tn* by

applying the results fromTheorem 2.4and6.20. Again, let*m*≥max{2*d*−1*, a/*2}and*d*≥42.

*dm*

*k*0

−1*dmk*

⎛

⎝2*d*−1*mk*−1

*dm*

⎞ ⎠

⎛
⎝*dm*

*k*

⎞

⎠_{T}_{k}_{}_{dm}_{−}_{γ}_{−}_{log}*a*

*b*

≤_{}*dm*
*k*0

−1*dmkgkSkdm*−*γ*−log

*a*
*b*

*dm*

*k*0

−1*dmkgkRkdm*

_{}*dm*

*k*0

−1*dmkgkSkdm*−*γ*−log

*a*
*b*

|*Em*|

*< c*4·

1−1*/dd*

*d*−14*d*
*m*

19

√

*πd*3
*d*−1*d*−2√*m*

2*dd*_{}_{d}_{−}_{1}_{}*d*−1
2*d*−12*d*−1

*m*

*.*

6.22

By similar arguments we get the same bound when*b > a*. For*d*≥3 it can easily be seen that

2*dd*_{}_{d}_{−}_{1}_{}*d*−1
2*d*−12*d*−1

22*d*−1

*d*−1 ·

1−1*/dd*

1−1*/*2*d*2*d* ·
1
4*d* *<*

18

4*d*1*.* 6.23

Thus, we finally have proven the following theorem.

**Theorem 6.2. Let**

*Tn*:
*an*−1

*j*1

1
*j* −

*bn*−1

*j*1

1
*j* 2

*n*−1

*j*1

1
*j* −

*n*_{}2_{−}_{1}

*j*1

1

*j,* *n >*1*,* 6.24

*where* *a, b* *are positive integers. Let* *d* ≥ *42 be an integer. Then, there is a positive constant* *c*5
*depending at most ona, banddsuch that*

*dm*

*k*0
−1*k*

2*d*−1*mk*−1
*dm*

*dm*

*k*

*Tkdm*−*γ*−log*a*

*b*

*<* √*cm*5

18
4*d*1

*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*

*c*6

*depending at most ona, banddsuch that for all integersm*≥

*1 one has*

*dm*

*k*0

−1*dmk*

2*d*−1*mk*−1
*dm*

*dm*

*k*

*Tkdm*−*γ*−log*a*

*b*

*< c*6·

1−1*/dd*

*d*−14*d*
*m*

*.*

7.1

A proof of this conjecture would be implied by suitable bounds for the integral stated
inLemma 5.1. For*j*1 such a bound follows from the double integral given inCorollary 5.2:

*dm*

*k*0

−1*dmkgk*

*dmk*2

1

0

1

0

1−*udm*1−*wmu*2*d*−1*m*−1* _{w}d*−1

*m*−1

1−1−*uwdm*1 *du dw*

1

0

1

0

1−*u*21−*wu*2

1−1−*uw*3

1−*uu*2_{w}

1−1−*uw*

*d*−2

×

1−*ud*1−*wu*2*d*−1* _{w}d*−1
1−1−

*uwd*

*m*−1

*du dw*

≤ 1

4*d*−2

1−1*/dd*

*d*−14*d*

m−1_{1}

0

_{1}

0

1−*u*21−*wu*2

1−1−*uw*3 *du dw*

2*d*−1

31−1*/dd* ·

1−1*/dd*

*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−*uu*2*w*
1−1−*uw,*

1−*ud*1−*wu*2*d*−1*wd*−1

1−1−*uwd* *,* 7.3

take their maximum values 42−*d*_{and}_{}_{1}_{−}_{1}_{/d}_{}*d _{/}*

_{}

_{d}_{−}

_{1}

_{}

_{4}

*d*

_{}

_{inside the unit square}

_{}

_{0}

_{,}_{1}

_{}

_{×}

_{}

_{0}

_{,}_{1}

_{}

at*u, w* 1*/*2*,*1and*u, w* 1*/*2*,*2*d*−2*/*2*d*−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

*T*1*d, m*:*c*4·

1−1*/dd*

*d*−14*d*
m

whereas we have inTheorem 6.2that

*T*2*d, m*: √*c*5
*m*

18
4*d*1

m

*,* *d*≥42*, m*≥1*.* 7.5

For fixed*d*≥42 and suﬃciently large*m*it is clear on the one hand that*T*1*d, m< T*2*d, m*,
but on the other hand we have

lim

*m*→ ∞*d*lim→ ∞

log*T*1*d, m*

log*T*2*d, m* 1*d*lim→ ∞*m*lim→ ∞

log*T*1*d, m*

log*T*2*d, m.* 7.6

Conversely, for*d*tending to infinity, one gets

−log|*T*1*d, m*| *dm*log 4*,* −log|*T*2*d, m*| *dm*log 4*,* 7.7

with implicit constants depending at most on*m*. For the denominators*bm*of the transformed

series *Skdm* in Theorem 2.4we have the bound log*bm* *d*2*m*2 from Theorem 4.1, and a

similar inequality holds for the denominators of the transformed series*Tkdm*inTheorem 6.2.

**References**

1 C. Elsner, “On a sequence transformation with integral coeﬃ*cients 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
coeﬃ*cients,” Mathematische Zeitschrift, vol. 255, no. 1, pp. 117–131, 2007.*

4 C. Elsner, “On a sequence transformation with integral coeﬃ*cients 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.