# On Rational Approximations to Euler's Constantγ and to γ+log(a/b)

Authors: Carsten Elsner
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

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 to γ and γ+logq (q∈ℚ>0) defined 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+logq. It is shown that for every q∈ℚ>0 and every integer d≥42 there are infinitely many rationals am/bm for m=1,2,… such that |γ+logq−am/bm|≪((1−1/d)d/(d−1)4d)m and bm∣Zm with logZm~12d2m2 for m tending to infinity.


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## Body

## 1. Introduction

Let(1.1)sn:=(11+12+13+⋯+1n-1)-logn(n≥2).
It is well known that the sequence (sn)n≥1 converges to Euler's constant γ=0,577…, where
(1.2)sn=γ+𝒪(1n)(n≥1).
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 author [1] introduced a linear transformation for the series (sn)n≥1 with integer coefficients which improves the rate of convergence. Let τ be an additional positive integer parameter.Proposition 1.1 (see [1]).
For any integersn≥1 and τ≥2 one has
(1.3)|∑k=0n(-1)n+k(n+k+τ-1n)(nk)·sk+τ-γ|≤(τ-1)!2n(n+1)(n+2)⋯(n+τ).
Particularly, by choosing τ=n≥2, one gets the following result.Corollary 1.2.
For any  integern≥2one has
(1.4)|∑k=0n(-1)n+k(2n+k-1n)(nk)·sn+k-γ|≤12n2(2nn)≤1n3/2·4n.Some authors have generalized the result of Proposition1.1 under various aspects. At first one cites a result due to Rivoal [2].Proposition (see [2]).
Forn tending to infinity, one has
(1.5)|γ-1(-2)n∑k=0n(-1)k(2n+2kn)(nk)s2k+n+1|=𝒪(1n27n/2).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 method [3].Proposition ([3]).
Forn tending to infinity the following asymptotic formula holds:
(1.6)|γ-∑k=0n(-1)n+k(n+kn)(nk)sk+n+1|=14n+o(n).Recently the author has found series transformations involving three parametersn, τ1 and τ2, [4]. In Propositions 1.5 and 1.6 certain integral representations of the (discrete) series transformations are given, which exhibit important (analytical) tools to estimate the error terms of the transformations.Proposition 1.5 (see [4]).
Letn≥1, τ1≥1, and τ2≥1 be integers. Additionally one assumes that
(1.7)1+τ1≤τ2.
Then one has
(1.8)∑k=0n(-1)n+k(n+τ1+kn)(nk)·sk+τ2-γ=(-1)n+1∫01(11-u+1logu)·uτ2-τ1-1·∂n∂un(un+τ1(1-u)nn!)du.Proposition (see [4]).
Letn≥1, τ1≥1 and τ2≥1 be integers. Additionally one assumes that
(1.9)1+τ1≤τ2≤1+n+τ1.
Then one has
(1.10)∑k=0n(-1)n+k(n+τ1+kn)(nk)·sk+τ2-γ=(-1)n+τ2-τ1∫01∫01w(t)·(1-u)n+τ1un(1-t)τ2-τ1-1tn+τ1-τ2+1(1-ut)n+1dudt,
with
(1.11)w(t):=1t·(π2+log2(1t-1)).Setting(1.12)n=τ2=dm,τ1=(d-1)m-1,(d≥2),
one gets an explicit upper bound from Proposition 1.6Corollary 1.7.
For integersm≥2, d≥3, one has
(1.13)|∑k=0dm(-1)dm+k((2d-1)m+k-1dm)(dmk)·sk+dm-γ|<Cd·((1-1/d)d(d-1)4d)m-2,
where 0<Cd≤1/16π2 is some constant depending only on d. For d=2 one gets
(1.14)|∑k=02m(-1)k(3m+k-12m)(2mk)·sk+2m-γ|<(167π)2·164m(m≥1).
For an application of Corollary 1.7 let the integers Bm and Am be defined by
(1.15)Bm:=l.c.m.(1,2,3,…,4m),Am:=Bm∑k=02m(-1)k(3m+k-12m)(2mk)·(1+12+⋯+1k+2m-1).Λ(k) denotes the von Mangoldt function. By [5, Theorem  434] one has
(1.16)ψ(m):=∑k≤mΛ(k)~m.
Then, for ε:=(log55)/4-1>0.0018, there is some integer m0 such that
(1.17)Bm=eψ(4m)<e4(1+ε)m=55m(m≥m0).
Multiplying (1.14) by Bm, we deduce the following corollary.Corollary.
There is an integerm0 such that one has for all integers m≥m0 that
(1.18)|Bm∑k=02m(-1)k(3m+k-12m)(2mk)·log(k+2m)+γBm-Am|<(167π)2·(5564)m.


## 2. Results on Rational Approximations toγ

In 2007, Aptekarev and his collaborators [6] found rational approximations to γ, which are based on a linear third-order recurrence. For the sake of brevity, let D(n)=l.c.m.(1,2,…,n).Proposition (see [6]).
Let(pn)n≥0 and (qn)n≥0 be two solutions of the linear recurrence
(2.1)(16n-15)(n+1)un+1=(128n3+40n2-82n-45)un-n(256n3-240n2+64n-7)un-1+n(n-1)(16n+1)un-2
with p0=0,  p1=2,  p2=31/2 and q0=1, q1=3, q2=25. Then, one has qn∈ℤ, D(n)pn∈ℤ, and
(2.2)|γ-pnqn|~c0e-22n,|qn|~c1n1/4(2n)!n!e2n,
with two positive constants c0,c1. It seems interesting to replace the fraction pn/qn by
(2.3)AnBn:=D(n)pnD(n)qn,
and to estimate the remainder in terms of Bn.Corollary 2.2.
Let0<ε<1. Then there are two positive constants c2,c3, such that for all sufficiently large integers n one has
(2.4)c2exp(-2(1+ε)2logBn/loglogBn)<|γ-AnBn|<c3exp(-2(1-ε)2logBn/loglogBn).Recently, Rivoal [7] presented a related approach to the theory of rational approximations to Euler's constant γ, and, more generally, to rational approximations for values of derivatives of the Gamma function. He studied simultaneous Padé approximants to Euler's functions, from which he constructed a third-order recurrence formula that can be applied to construct a sequence in ℚ(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 (see [7]).
(i) The recurrence(2.5)(n+3)2(8n+11)(8n+19)Un+3=(24n2+145n+215)(8n+11)Un+2-(24n3+105n2+124n+25)(8n+27)Un+1+(n+2)2(8n+19)(8n+27)Un,
provides two sequences of rational numbers (pn)n≥0 and (qn)n≥0 with p0=-1, p1=4, p2=77/4 and q0=1, q1=7, q2=65/2 such that (pn/qn)n≥0 converges to γ.(ii)  The recurrence
(2.6)(n+1)(n+2)(n+3)Un+3=(3n2+19n+29)(n+1)Un+2-(3n3+6n2-7n-13)Un+1+(n+2)3Un,
provides two sequences of rational numbers (pn)n≥0 and (qn)n≥0 with p0=-1, p1=11, p2=71 and q0=0, q1=8, q2=56 such that (pn/qn)n≥0 converges to log(2)+γ.The goal of this paper is to construct rational approximations toγ+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 convergence (see Theorems 2.4 and 6.2 below).In the following we denote byB2n the Bernoulli numbers, that is,  B2=1/6, B4=-1/30, B6=1/42, and so on (In Sections 3–6 the Bernoulli numbers cannot be confused with the integers Bn from Corollary 2.2.) In this paper we will prove the following result.Theorem 2.4.
Leta≥1,  b≥1,  d≥42 and  m≥1 be positive integers, and
(2.7)Sn:=∑j=1an-11j-∑j=1bn-11j+2∑j=1n-11j-∑j=1n2-11j-12n2+∑j=1dmB2j2j(1n2j(1a2j-1b2j+1)-1n4j),(n≥1).
Then,
(2.8)|∑k=0dm(-1)dm+k((2d-1)m+k-1dm)(dmk)Sk+dm-γ-logab|<c4·((1-1/d)d(d-1)4d)m,
where c4 is some positive constant depending only on d.


## 3. Proof of Theorem2.4

Lemma 3.1.
One has for positive integersd and m(3.1)g(k)≔((2d-1)m+k-1dm)(dmk)<16dm(0≤k≤dm).Proof.
Applying the well known inequality(gh)≤2g, we get
(3.2)((2d-1)m+k-1dm)(dmk)≤2(2d-1)m+dm-12dm=24dm-m-1<16dm.
This proves the lemma.g(k) takes its maximum value for k=k0 with
(3.3)k0=5d2-4d+1-d+12m+𝒪(1),
which leads to a better bound than 16dm in Lemma 3.1. But we are satisfied with Lemma 3.1. A main tool in proving Theorem 2.4 is Euler's summation formula in the form
(3.4)∑i=1nf(i)=∫1nf(x)dx+f(1)+f(n)2+∑j=1rB2j(2j)!(f(2j-1)(n)-f(2j-1)(1))+Rr,
where r∈ℕ is a suitable chosen parameter, and the remainder Rr is defined by a periodic Bernoulli polynomial P2r+1(x), namely
(3.5)Rr=1(2r+1)!∫1nP2r+1(x)f(2r+1)(x)dx,
with
(3.6)P2r+1(x)=(-1)r-1(2r+1)!∑j=1∞2sin(2πjx)(2πj)2r+1.
Applying the summation formula to the function f(x)=1/x, we get (see [8, equation ( 5)] )
(3.7)∑i=1n-11i=logn+12-12n+∑j=1rB2j2j(1-1n2j)-∫1nP2r+1(x)x2r+2dx,(n,r∈ℕ).
It follows that
(3.8)∑i=nn2-11i-logn=12n-12n2+∑j=1rB2j2j(1n2j-1n4j)-∫nn2P2r+1(x)x2r+2dx,(n,r∈ℕ).
We prove Theorem 2.4 for a≥b. The case a<b is treated similarly. So we have again by the above summation formula that
(3.9)∑i=bnan-11i-logab=(1b-1a)12n+∑j=1rB2j2jn2j(1b2j-1a2j)-∫bnanP2r+1(x)x2r+2dx,(n,r∈ℕ).
First, we estimate the integral on the right-hand side of (3.8). We have
(3.10)|∫nn2P2r+1(x)x2r+2dx|≤∫nn2|P2r+1(x)|x2r+2dx≤∫n∞|P2r+1(x)|x2r+2dx≤2(2r+1)!∫n∞1x2r+2∑j=1∞1(2πj)2r+1dx=2(2r+1)!(2π)2r+1[-1(2r+1)x2r+1]x=n∞∑j=1∞1j2r+1=2(2r)!(2π)2r+1n2r+1ζ(2r+1)<3(2r)!(2π)2r+1n2r+1,
since 2ζ(2r+1)≤2ζ(3)<3. Next, we assume that n≥a. Hence [bn,an]⊆[n,n2], and therefore we estimate the integral on the right-hand side in (3.9) by
(3.11)|∫bnanP2r+1(x)x2r+2dx|≤∫bnan|P2r+1(x)|x2r+2dx≤∫nn2|P2r+1(x)|x2r+2dx≤3(2r)!(2π)2r+1n2r+1.
In the sequel we put r=dm. Moreover, in the above formula we now replace n by dm+k with 0≤k≤dm. In order to estimate (2r)! we use Stirling's formula
(3.12)2πm(me)m<m!<2π(m+1)(me)m,(m>0).
Then, it follows that
(3.13)|∫dm+k(dm+k)2P2r+1(x)x2r+2dx|≤3(2r)!(2π)2r+1(dm+k)2r+1≤3(2r)!(2π)2r+1(dm)2r+1=3(2dm)!(2π)2dm+1(dm)2dm+1≤3π(2dm+1)(2πdm)2dm+1·(2dme)2dm≤33πdm(2πdm)(πe)2dm,
and similarly we have
(3.14)|∫b(dm+k)a(dm+k)P2r+1(x)x2r+2dx|≤33πdm(2πdm)(πe)2dm,(dm≥a).
By using the definition of Sn in Theorem 2.4, the formula (1.1) for sn, and the identities (3.8), (3.9), it follows that
(3.15)Sn-γ-logab=(sn-γ)+(sn-sn2)+(san-sbn)-12n2+∑j=1rB2j2j(1n2j(1a2j-1b2j+1)-1n4j)=(sn-γ)+12n(1b-1a-1)+∫nn2P2r+1(x)x2r+2dx-∫bnanP2r+1(x)x2r+2dx,
where r is specified to r=dm and n to n=dm+k. Moreover, we know from [4, Lemma  2] that
(3.16)∑k=0dm(-1)dm+kg(k)=1,(m≥1).
By setting n=dm+k, the above formula for the series transformation of Sdm+k simplifies to
(3.17)|∑k=0dm(-1)dm+kg(k)Sdm+k-γ-logab|=|∑k=0dm(-1)dm+kg(k)(sdm+k-γ)+12(1b-1a-1)∑k=0dm(-1)dm+kg(k)dm+k+∑k=0dm(-1)dm+kg(k)∫dm+k(dm+k)2P2r+1(x)x2r+2dx-∑k=0dm(-1)dm+kg(k)∫b(dm+k)a(dm+k)P2r+1(x)x2r+2dx|≤|∑k=0dm(-1)dm+kg(k)(sdm+k-γ)|+∑k=0dmg(k)|∫dm+k(dm+k)2P2r+1(x)x2r+2dx|+∑k=0dmg(k)|∫b(dm+k)a(dm+k)P2r+1(x)x2r+2dx|<Cd·((1-1/d)d(d-1)4d)m-2+∑k=0dmg(k)33πdmπdm(πe)2dm,
where dm≥a,m≥2, and d≥3. Here, we have used the results from Corollary 1.7, (3.13), and (3.14). The sum
(3.18)∑k=0dm(-1)dm+kg(k)dm+k
vanishes, since for every real number x>-dm we have
(3.19)∑k=0dm(-1)dm+k((2d-1)m+k-1dm)(dmk)dm+k+x=(1-(d-1)m+x)⋯(m+x)(dm+x)dm+1,
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(3.20)(64(πe)2)d<(1-1/d)d2(d-1)
holds for all integers d≥42. Now, using Lemma 3.1, we estimate the right-hand side in (3.17) for dm≥a and d≥42 as follows:
(3.21)|∑k=0dm(-1)dm+kg(k)Sdm+k-γ-logab|<Cd·((1-1/d)d(d-1)4d)m-2+∑k=0dm33πdmπdm16dm(πe)2dm=Cd·((1-1/d)d(d-1)4d)m-2+3(dm+1)3dmdmπ14dm(64(πe)2)dm<(3.20)Cd·((1-1/d)d(d-1)4d)m-2+3(dm+1)3dmdm2mπ((1-1/d)d(d-1)4d)m≤Cd·((1-1/d)d(d-1)4d)m-2+85283dπ((1-1/d)d(d-1)4d)m≤c4((1-1/d)d(d-1)4d)m.
The last but one estimate holds for all integers m≥2, d≥42, and c4 is a suitable positive real constant depending on d. This completes the proof of Theorem 2.4.


## 4. On the Denominators ofSn

In this section we will investigate the size of the denominatorsbm of our series transformations
(4.1)ambm=∑k=0dm(-1)dm+kg(k)Sk+dm,
for m tending to infinity, where am∈ℤ and bm∈ℕ are coprime integers.Theorem 4.1.
For everym≥1 there is an integer  Zm with  Zm>0,  bm|Zm, and
(4.2)logZm~12d2m2,(m→∞).Proof.
We will need some basic facts on the arithmetical functionsϑ(x) and ψ(x). Let
(4.3)ϑ(x)=∑p≤xlogp,(x>1),ψ(x)=∑p≤x[logxlogp]logp,(x>1),
where p is restricted on primes. Moreover, let Dn:=l.c.m(1,2,…,n) for positive integers n. Then,
(4.4)ψ(n)=logDn,(n≥1),(4.5)ψ(x)~ϑ(x)~π(x)logx~x,(x→∞),
where (4.5) follows from [5, Theorem  420] and the prime number theorem. By [5, Theorem  118] (von Staudt's theorem) we 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 primes p with (p-1)∣2k. Hence,
(4.6)B2k∏p≤2k+1p∈ℤ,(k=1,2,…).
Next, let max{a,b}≤dm≤n≤2dm (n=k+dm are the subscripts of Sk+dm in Theorem 2.4). First, we consider the following terms from the series transformation in Sm:
(4.7)∑j=1an-11j-∑j=1bn-11j+2∑j=1n-11j-∑j=1n2-11j=:∑j=1n2-1ejj,
with
(4.8)ej:={1,if1≤j≤n-1-1,ifn≤j≤bn-10,ifbn≤j≤an-1-1,ifan≤j≤n2-1(a≥b),ej:={1,if1≤j≤n-1-1,ifn≤j≤an-1-2,ifan≤j≤bn-1-1,ifbn≤j≤n2-1.(a<b).
For every m≥1 there is a rational xm/ym defined by
(4.9)xmym=∑k=0dm(-1)dm+kg(k)∑j=1(k+dm)2-1ejj,
where xm∈ℤ, ym∈ℕ, (xm,ym)=1, and
(4.10)ym∣Ym:=D4d2m2,(dm≥max{a,b}).
Similarly, we define rationals um/vm by(4.11)umvm=∑k=0dm(-1)dm+kg(k)×(-12(k+dm)2+∑j=1dmB2j2j(1(k+dm)2j(1a2j-1b2j+1)-1(k+dm)4j)),
where um∈ℤ, vm∈ℕ and (um,vm)=1. We have
(4.12)(k+dm)2j∣(k+dm)4dm,(0≤k≤dm,1≤j≤dm).
Therefore, using the conclusion (4.6) from von Staudt's theorem, we get
(4.13)vm∣Vm:=2(ab)2dmDdm(∏p≤2dm+1p)(D2dm)4dm,(dm≥max{a,b}).
Note that D2dm=l.c.m.(dm,…,2dm), since every integer n1 with 1≤n1<dm divides at least one integer n2 with dm≤n2≤2dm.
From (4.10) and (4.13) we conclude on
(4.14)bm∣Zm:=2(ab)2dmDdmD4d2m2(D2dm)4dm(∏p≤2dm+1p).
Hence we have from (4.4) and (4.5) that
(4.15)logZm=log2+2dmlog(ab)+ψ(dm)+ψ(4d2m2)+4dmψ(2dm)+ϑ(2dm+1)~log2+2dmlog(ab)+dm+4d2m2+8d2m2+(2dm+1)=1+log2+(3+2log(ab))dm+12d2m2~12d2m2(m→∞).
The theorem is proved.Remark 4.2.
On the one side we have shown thatlogYm~4d2m2 and logVm~8d2m2. On the other side, every prime p dividing Vm satisfies p≤max{a,b,dm,2dm+1,2dm}=2dm+1 and therefore p divides Ym=D4d2m2. Conversely, all primes p with 2dm+1<p<4d2m2 divide Ym, but not Vm. That means:  Vm is much bigger than Ym, but Vm is formed by powers of small primes, whereas Ymis divisible by many big primes.


## 5. Simplification of the Transformed Series

Let(5.1)Rn:=-12n2+∑j=1dmB2j2j(1n2j(1a2j-1b2j+1)-1n4j),
such that
(5.2)Sn=∑j=1an-11j-∑j=1bn-11j+2∑j=1n-11j-∑j=1n2-11j+Rn.
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 Rk+dm to the series transformation in Theorem 2.4. For this purpose, we define
(5.3)Em:=∑k=0dm(-1)dm+k((2d-1)m+k-1dm)(dmk)Rk+dm=-12∑k=0dm(-1)dm+kg(k)(dm+k)2+∑j=1dmB2j2j((1a2j-1b2j+1)∑k=0dm(-1)dm+kg(k)(dm+k)2j-∑k=0dm(-1)dm+kg(k)(dm+k)4j).
A major step in estimating Em is to expressthe sums on the right-hand side by integrals.Lemma 5.1.
For positive integersd,j and m one has
(5.4)∑k=0dm(-1)dm+kg(k)(dm+k)2j=-(-1)dm(dm)!(2j-1)!∫01um(logu)2j-1∂dm∂udm(u(2d-1)m-1(1-u)dm)du.Proof.
For integersk,r and a real number ρ with k+ρ>0 the identity
(5.5)1(k+ρ)r=1(r-1)!∫0∞e-(k+ρ)ttr-1dt
holds, which we apply with r=2j and ρ=dm to substitute the fraction 1/(dm+k)2j. Introducing the new variable u:=e-t, we then get
(5.6)∑k=02m(-1)dm+kg(k)(k+dm)2j=-(-1)dm(2j-1)!∑k=0dm(-1)kg(k)∫01uk+dm-1(logu)2j-1du=-(-1)dm(2j-1)!∫01(∑k=0dm(-1)kg(k)uk+(d-1)m-1)um(logu)2j-1du.
The sum inside the brackets of the integrand can be expressed by using the equation
(5.7)∑k=0n(-1)k(n+τ+kn)(nk)uτ+k=∂n∂un(un+τ(1-u)nn!),(n,τ∈ℕ∪{0}),
in which we put n=dm and τ=(d-1)m-1. This gives the identity stated in the lemma.The following result deals with the casej=1, in which we express the finite sum by a double integral on a rational function.Corollary.
For every positive integerm one has
(5.8)∑k=0dm(-1)dm+kg(k)(dm+k)2=(-1)(d-1)m∫01∫01(1-u)dm(1-w)mu(2d-1)m-1w(d-1)m-1(1-(1-u)w)dm+1dudw.Proof.
Setj=1 in Lemma 5.1, and note that
(5.9)logu=-(1-u)∫01dw1-(1-u)w.
Hence,
(5.10)∑k=0dm(-1)dm+kg(k)(dm+k)2=-(-1)dm(dm)!∫01umlogu∂dm∂udm(u(2d-1)m-1(1-u)dm)du=(-1)dm(dm)!∫01∫01(1-u)um1-(1-u)w∂dm∂udm(u(2d-1)m-1(1-u)dm)dudw.
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:
(5.11)us1-(1-u)w=∑ν=0s-1(w-1)νwν+1us-ν-1+(w-1w)s11-(1-u)w.
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:
(5.12)∂dm∂udm(us1-(1-u)w)=(w-1w)s(-1)dm(dm)!wdm(1-(1-u)w)dm+1.
Therefore, we get from (5.10) by iterated integrations by parts:
(5.13)∑k=0dm(-1)dm+kg(k)(dm+k)2=1(dm)!∫01∫01u(2d-1)m-1(1-u)dm∂dm∂udm(um-um+11-(1-u)w)dudw=1(dm)!∫01∫01u(2d-1)m-1(1-u)dm((w-1w)m-(w-1w)m+1)×(-1)dm(dm)!wdmdudw(1-(1-u)w)dm+1.
The corollary is proved by noting that
(5.14)(w-1w)m-(w-1w)m+1=(-1)m(1-w)mwm+1.


## 6. EstimatingEm

In this section we estimateEm defined in (5.3). Substituting 1-u for u into the integral in Lemma 5.1 and applying iterated integration by parts, we get
(6.1)∑k=0dm(-1)dm+kg(k)(dm+k)2j=-(-1)dm(dm)!(2j-1)!∫01∂dm∂udm((1-u)m(log(1-u))2j-1)((1-u)(2d-1)m-1udm)du.
Set
(6.2)f(u):=(1-u)m(log(1-u))2j-1,
where m and j are kept fixed. We have f(0)=0. For an integer k>0 we use Cauchy's formula
(6.3)f(k)(a)=k!2πi∫Cf(z)(z-a)k+1dz
to estimate |f(k)(0)|. Let C denote the circle in the complex plane centered around 0 with radius R:=1-1/2k. With a=0 and f(z) defined above, Cauchy's formula yields the identity
(6.4)f(k)(0)=k!2πRk∫-ππe-ikϕ(1-Reiϕ)mlog2j-1(1-Reiϕ)dϕ.
For the complex logarithm function occurring in (6.4) we 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∉(-∞,0] are taken from the interval (-π,π). Therefore, using 1-Reiϕ=1-Rcosϕ-iRsinϕ, we get
(6.5)|1-Reiϕ|=1+R2-2Rcosϕ=:A(R,ϕ),arg(1-Reiϕ)=-arctan(Rsinϕ1-Rcosϕ).
Hence,
(6.6)log(1-Reiϕ)=ln1+R2-2Rcosϕ-iarctan(Rsinϕ1-Rcosϕ)=12ln(A(R,ϕ))-iarctan(Rsinϕ1-Rcosϕ).
Thus, it follows from (6.4) that(6.7)|f(k)(0)|≤k!2πRk∫-ππ|1-Reiϕ|m·|log(1-Reiϕ)|2j-1dϕ=k!2πRk∫-ππ(A(R,ϕ))m/2(14ln2(A(R,ϕ))+arctan2(Rsinϕ1-Rcosϕ))(2j-1)/2dϕ=k!πRk∫0π(A(R,ϕ))m/2(14ln2(A(R,ϕ))+arctan2(Rsinϕ1-Rcosϕ))(2j-1)/2dϕ.
From 0<R<1 we conclude on
(6.8)0<(1-R)2=1+R2-2R≤1+R2-2Rcosϕ=A(R,ϕ)<4,(0≤ϕ≤π),0≤Rsinϕ1-Rcosϕ≤sinϕ1-cosϕ,(0<ϕ≤π).
Since arctan is a strictly increasing function, we get
(6.9)arctan(Rsinϕ1-Rcosϕ)≤arctan(sinϕ1-cosϕ)=arctan cot(ϕ2)=arctan(tan(π-ϕ2))=π-ϕ2,(0<ϕ≤π).
For 0<R<1, this upper bound also holds for ϕ=0. Finally, we note that Rk=(1-1/2k)k≥1/2. Altogether, we conclude from (6.7) on
(6.10)|f(k)(0)|≤k!πRk∫0π4m/2(ln244+arctan2(sinϕ1-cosϕ))(2j-1)/2dϕ≤2m+1k!π∫0π(ln22+(π-ϕ2)2)(2j-1)/2dϕ≤2m+1k!π∫0π(ln22+π24)j-1/2dϕ≤2m+1k!π∫0π3j-1/2dϕ≤2m+13jk!.
It follows that the Taylor series expansion of f(u),
(6.11)f(u)=∑k=0∞f(k)(0)k!uk,
converges at least for -1<u<1. Then,
(6.12)f(dm)(u)=∑k=dm∞f(k)(0)(k-dm)!uk-dm=∑k=0∞f(k+dm)(0)k!uk,
and the estimate given by (6.10) implies for 0<u<1 that
(6.13)|f(dm)(u)|≤∑k=0∞|f(k+dm)(0)|k!uk≤2m+13j∑k=0∞(k+dm)!k!uk=2m+13j(dm)!∑k=0∞(k+dmk)uk=2m+13j(dm)!(1-u)dm+1.
Combining (6.13) with the result from (6.1), we get for m>1(6.14)|∑k=0dm(-1)dm+kg(k)(dm+k)2j|≤2m+13j(2j-1)!∫01(1-u)(d-1)m-2udmdu=2m+13j(2j-1)!Γ(dm+1)Γ((d-1)m-1)Γ((2d-1)m)=2d-1d-1·2m+13j(2j-1)!((d-1)m-1)·1((2d-1)mdm).We estimate the binomial coefficient by Stirling's formula (3.12). For this purpose we additionally assume that m≥2d-1:
(6.15)((2d-1)mdm)≥(2d-1)m2π(dm+1)((d-1)m+1)((2d-1)2d-1dd(d-1)d-1)m≥2d-12πd2m((2d-1)2d-1dd(d-1)d-1)m.
We now assume m≥2d-1 and substitute the above inequality into (6.14):
(6.16)|∑k=0dm(-1)dm+kg(k)(dm+k)2j|≤d(2d-1)2m+13j2πm(2j-1)!(d-1)((d-1)m-1)2d-1(dd(d-1)d-1(2d-1)2d-1)m.
For all integers m≥1 and d≥1 we have
(6.17)(2d-1)2πm2d-1<2πdm,(d-1)((d-1)m-1)≥(d-1)(d-2)m.
Thus we have proven the following result.Lemma 6.1.
For all integersd,m with d≥3 and m≥2d-1 one has
(6.18)|∑k=0dm(-1)dm+kg(k)(dm+k)2j|<2m+23jπd3(2j-1)!(d-1)(d-2)m(dd(d-1)d-1(2d-1)2d-1)m.Next, we need an upper bound for the Bernoulli numbersB2j (cf. [9, 23.1.15] ):
(6.19)|B2j|≤2(2j)!(2π)2j(1-21-2j)≤4(2j)!(2π)2j,(j≥1).
Let d≥3 and m≥max{2d-1,a/2}. Using this and Lemma 6.1, we estimate Em in (5.3):
(6.20)|Em|<32πd32m+2(d-1)(d-2)m(dd(d-1)d-1(2d-1)2d-1)m+πd32m+2(d-1)(d-2)m(dd(d-1)d-1(2d-1)2d-1)m×∑j=1dmB2j2j(|1a2j-1b2j+1|3j(2j-1)!+32j(4j-1)!)≤πd32m+2(d-1)(d-2)m(dd(d-1)d-1(2d-1)2d-1)m×(32+∑j=1dm4(2j-1)!(2π)2j(2·3j(2j-1)!+32j(4j-1)!))<4πd3(d-1)(d-2)m(2dd(d-1)d-1(2d-1)2d-1)m(32+8∑j=1∞((32π)2j+(32π)2j))<19πd3(d-1)(d-2)m(2dd(d-1)d-1(2d-1)2d-1)m.
Now, let
(6.21)Tn:=∑j=1an-11j-∑j=1bn-11j+2∑j=1n-11j-∑j=1n2-11j=∑j=1n2-1ejj,(n>1),
with the numbers ej introduced in the proof of Theorem 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 from Theorem 2.4 and (6.20). Again, let m≥max{2d-1,a/2} and d≥42.
(6.22)|∑k=0dm(-1)dm+k((2d-1)m+k-1dm)(dmk)Tk+dm-γ-logab|≤|∑k=0dm(-1)dm+kg(k)Sk+dm-γ-logab|+|∑k=0dm(-1)dm+kg(k)Rk+dm|=|∑k=0dm(-1)dm+kg(k)Sk+dm-γ-logab|+|Em|<c4·((1-1/d)d(d-1)4d)m+19πd3(d-1)(d-2)m(2dd(d-1)d-1(2d-1)2d-1)m.
By similar arguments we get the same bound when b>a. For d≥3 it can easily be seen that
(6.23)2dd(d-1)d-1(2d-1)2d-1=2(2d-1)d-1·(1-1/d)d(1-1/2d)2d·14d<184d+1.
Thus, we finally have proven the following theorem.Theorem 6.2.
Let(6.24)Tn:=∑j=1an-11j-∑j=1bn-11j+2∑j=1n-11j-∑j=1n2-11j,(n>1),
where a,b are positive integers. Let d≥42 be an integer. Then, there is a positive constant c5 depending at most on a,b and d such that
(6.25)|∑k=0dm(-1)k((2d-1)m+k-1dm)(dmk)Tk+dm-γ-logab|<c5m(184d+1)m,(m≥1).


## 7. Concluding Remarks

It seems that in Theorem6.2 a smaller bound holds.Conjecture.
Leta,b be positive integers.  Let d≥2 be an integer.  Then there is a positive constant c6 depending at most on a,b and d such that for all integers m≥1 one has
(7.1)|∑k=0dm‍(-1)dm+k((2d-1)m+k-1dm)(dmk)Tk+dm-γ-logab|<c6·((1-1/d)d(d-1)4d)m.A proof of this conjecture would be implied by suitable bounds for the integral stated in Lemma5.1. For j=1 such a bound follows from the double integral given in Corollary 5.2:
(7.2)|∑k=0dm‍(-1)dm+kg(k)(dm+k)2|=∫01∫01(1-u)dm(1-w)mu(2d-1)m-1w(d-1)m-1(1-(1-u)w)dm+1dudw=∫01∫01(1-u)2(1-w)u21-(1-u)w3((1-u)u2w1-(1-u)w)d-2×((1-u)d(1-w)u2d-1wd-1(1-(1-u)w)d)m-1dudw≤14d-2((1-1/d)d(d-1)4d)m-1∫01∫01(1-u)2(1-w)u21-(1-u)w3dudw=2(d-1)3(1-1/d)d·((1-1/d)d(d-1)4d)m,(m≥1),
where the double integral in the last but one line equals to 1/24.Note that the rational functions(7.3)(1-u)u2w1-(1-u)w,(1-u)d(1-w)u2d-1wd-1(1-(1-u)w)d,
take their maximum values 42-d and (1-1/d)d/((d-1)4d) inside the unit square [0,1]×[0,1] at (u,w)=(1/2,1) and (u,w)=(1/2,(2d-2)/(2d-1)), respectively. Finally, we compare the bound for the series transformation given by Theorem 2.4 with the bound proven for Theorem 6.2. In Theorem 2.4 the bound is
(7.4)T1(d,m):=c4·((1-1/d)d(d-1)4d)m,(d≥42,m≥1),
whereas we have in Theorem 6.2 that
(7.5)T2(d,m):=c5m(184d+1)m,(d≥42,m≥1).
For fixed d≥42 and sufficiently large m it is clear on the one hand that T1(d,m)<T2(d,m), but on the other hand we have
(7.6)limm→∞limd→∞logT1(d,m)logT2(d,m)=1=limd→∞limm→∞logT1(d,m)logT2(d,m).
Conversely, for d tending to infinity, one gets
(7.7)-log|T1(d,m)|≫dmlog4,-log|T2(d,m)|≫dmlog4,
with implicit constants depending at most on m. For the denominators bm of the transformed series Sk+dm in Theorem 2.4 we have the bound logbm≪d2m2 from Theorem 4.1, and a similar inequality holds for the denominators of the transformed series Tk+dm in Theorem 6.2.



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*Source: 626489-2009-06-28.xml*