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

Abstract

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+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 𝑎_𝑚/𝑏_𝑚 for 𝑚=1,2,… such that |𝛾+log𝑞−𝑎_𝑚/𝑏_𝑚|≪((1−1/𝑑)^𝑑/(𝑑−1)4^𝑑)^𝑚 and 𝑏_𝑚∣𝑍_𝑚 with log𝑍_𝑚∼12𝑑²𝑚² for 𝑚 tending to infinity.

1. Introduction

Let 𝑠_𝑛1∶=(1+12+131+⋯+𝑛−1)−log𝑛(𝑛≥2). (1.1) It is well known that the sequence (𝑠_𝑛)_𝑛≥1 converges to Euler's constant 𝛾=0,577…, where 𝑠_𝑛1=𝛾+𝒪(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 author [1] introduced a linear transformation for the series (𝑠_𝑛)_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 integers 𝑛≥1 and 𝜏≥2 one has |^𝑛_𝑘=0(−1)^𝑛+𝑘𝑛𝑛𝑘𝑛+𝑘+𝜏−1⋅𝑠_𝑘+𝜏−𝛾|≤(𝜏−1)!.2𝑛(𝑛+1)(𝑛+2)⋯(𝑛+𝜏) (1.3) Particularly, by choosing 𝜏=𝑛≥2, one gets the following result.

Corollary 1.2. For any integer 𝑛≥2one has |^𝑛_𝑘=0(−1)^𝑛+𝑘𝑛𝑛𝑘2𝑛+𝑘−1⋅𝑠_𝑛+𝑘1−𝛾|≤2𝑛²(_𝑛^2𝑛)≤1𝑛^3/2⋅4^𝑛. (1.4)

Some authors have generalized the result of Proposition 1.1 under various aspects. At first one cites a result due to Rivoal [2].

Proposition (see [2]). For 𝑛 tending to infinity, one has 1|𝛾−(−2)^𝑛𝑛_𝑘=0(−1)^𝑘𝑛𝑛𝑘𝑠2𝑛+2𝑘_2𝑘+𝑛+11|=𝒪(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 method [3].

Proposition ([3]). For 𝑛 tending to infinity the following asymptotic formula holds: |𝛾−^𝑛_𝑘=0(−1)^𝑛+𝑘𝑛𝑛𝑘𝑠𝑛+𝑘_𝑘+𝑛+11|=4^{𝑛+𝑜(𝑛)}. (1.6)

Recently the author has found series transformations involving three parameters 𝑛, 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]). Let 𝑛≥1, 𝜏₁≥1, and 𝜏₂≥1 be integers. Additionally one assumes that 1+𝜏₁≤𝜏₂. (1.7) Then one has ^𝑛_𝑘=0(−1)^𝑛+𝑘𝑛+𝜏₁𝑛𝑛𝑘+𝑘⋅𝑠_𝑘+𝜏2−𝛾=(−1)^𝑛+11₀(1+11−𝑢𝑙𝑜𝑔𝑢)⋅𝑢^{𝜏2−𝜏1−1}⋅𝜕^𝑛𝜕𝑢^𝑛(𝑢^𝑛+𝜏1(1−𝑢)^𝑛𝑛!)𝑑𝑢. (1.8)

Proposition (see [4]). Let 𝑛≥1, 𝜏₁≥1 and 𝜏₂≥1 be integers. Additionally one assumes that 1+𝜏₁≤𝜏₂≤1+𝑛+𝜏₁. (1.9) Then one has ^𝑛_𝑘=0(−1)^𝑛+𝑘𝑛+𝜏₁𝑛𝑛𝑘+𝑘⋅𝑠_𝑘+𝜏2−𝛾=(−1)^{𝑛+𝜏2−𝜏11}₀¹₀𝑤(𝑡)⋅(1−𝑢)^𝑛+𝜏1𝑢^𝑛(1−𝑡)^{𝜏2−𝜏1−1}𝑡^{𝑛+𝜏1−𝜏2+1}(1−𝑢𝑡)^𝑛+1𝑑𝑢𝑑𝑡, (1.10) with 1𝑤(𝑡)∶=𝑡⋅(𝜋²+log²(1𝑡.−1)) (1.11)

Setting 𝑛=𝜏₂=𝑑𝑚,𝜏₁=(𝑑−1)𝑚−1,(𝑑≥2), (1.12) one gets an explicit upper bound from Proposition 1.6

Corollary 1.7. For integers 𝑚≥2, 𝑑≥3, one has |^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑘(2𝑑−1)𝑚+𝑘−1𝑑𝑚𝑑𝑚⋅𝑠_{𝑘+𝑑𝑚}−𝛾|<𝐶_𝑑⋅((1−1/𝑑)^𝑑(𝑑−1)4^𝑑)^𝑚−2, (1.13) where 0<𝐶_𝑑≤1/16𝜋² is some constant depending only on 𝑑. For 𝑑=2 one gets |^2𝑚_𝑘=0(−1)^𝑘𝑘3𝑚+𝑘−12𝑚2𝑚⋅𝑠_𝑘+2𝑚−𝛾|<(16)7𝜋²⋅164^𝑚(𝑚≥1). (1.14) For an application of Corollary 1.7 let the integers 𝐵_𝑚 and 𝐴_𝑚 be defined by 𝐵_𝑚𝐴∶=𝑙.𝑐.𝑚.(1,2,3,…,4𝑚),_𝑚∶=𝐵_𝑚^2𝑚_𝑘=0(−1)^𝑘𝑘3𝑚+𝑘−112𝑚2𝑚⋅(1+21+⋯+𝑘+2𝑚−1). (1.15)Λ(𝑘) denotes the von Mangoldt function. By [5, Theorem  434] one has 𝜓(𝑚)∶=_𝑘≤𝑚Λ(𝑘)∼𝑚. (1.16) Then, for 𝜀∶=(log55)/4−1>0.0018, there is some integer 𝑚₀ such that 𝐵_𝑚=𝑒^𝜓(4𝑚)<𝑒^{4(1+𝜀)𝑚}=55^𝑚(𝑚≥𝑚₀). (1.17) Multiplying (1.14) by 𝐵_𝑚, we deduce the following corollary.

Corollary. There is an integer 𝑚₀ such that one has for all integers 𝑚≥𝑚₀ that |𝐵_𝑚^2𝑚_𝑘=0(−1)^𝑘𝑘3𝑚+𝑘−12𝑚2𝑚⋅log(𝑘+2𝑚)+𝛾𝐵_𝑚−𝐴_𝑚|<(16)7𝜋²⋅(55)64^𝑚. (1.18)

2. Results on Rational Approximations to 𝜸

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 (𝑝_𝑛)_𝑛≥0 and (𝑞_𝑛)_𝑛≥0 be two solutions of the linear recurrence (16𝑛−15)(𝑛+1)𝑢_𝑛+1=(128𝑛³+40𝑛²−82𝑛−45)𝑢_𝑛−𝑛(256𝑛³−240𝑛²+64𝑛−7)𝑢_𝑛−1+𝑛(𝑛−1)(16𝑛+1)𝑢_𝑛−2 (2.1) with 𝑝₀=0, 𝑝₁=2, 𝑝₂=31/2 and 𝑞₀=1, 𝑞₁=3, 𝑞₂=25. Then, one has 𝑞_𝑛∈ℤ, 𝐷(𝑛)𝑝_𝑛∈ℤ, and 𝑝|𝛾−^𝑛𝑞_𝑛|∼𝑐₀𝑒^√−22𝑛,|𝑞_𝑛𝑐|∼¹𝑛^1/4(2𝑛)!𝑒𝑛!^√2𝑛, (2.2) with two positive constants 𝑐₀,𝑐₁. It seems interesting to replace the fraction 𝑝_𝑛/𝑞_𝑛 by 𝐴_𝑛𝐵_𝑛∶=𝐷(𝑛)𝑝_𝑛𝐷(𝑛)𝑞_𝑛, (2.3) and to estimate the remainder in terms of 𝐵_𝑛.

Corollary 2.2. Let 0<𝜀<1. Then there are two positive constants 𝑐₂,𝑐₃, such that for all sufficiently large integers 𝑛 one has 𝑐₂exp(−2(1+𝜀)√2log𝐵_𝑛/loglog𝐵_𝑛)𝐴<|𝛾−^𝑛𝐵_𝑛|<𝑐₃exp(−2(1−𝜀)√2log𝐵_𝑛/loglog𝐵_𝑛). (2.4)

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 (𝑛+3)²(8𝑛+11)(8𝑛+19)𝑈_𝑛+3=(24𝑛²+145𝑛+215)(8𝑛+11)𝑈_𝑛+2−(24𝑛³+105𝑛²+124𝑛+25)(8𝑛+27)𝑈_𝑛+1+(𝑛+2)²(8𝑛+19)(8𝑛+27)𝑈_𝑛, (2.5) provides two sequences of rational numbers (𝑝_𝑛)_𝑛≥0 and (𝑞_𝑛)_𝑛≥0 with 𝑝₀=−1, 𝑝₁=4, 𝑝₂=77/4 and 𝑞₀=1, 𝑞₁=7, 𝑞₂=65/2 such that (𝑝_𝑛/𝑞_𝑛)_𝑛≥0 converges to 𝛾.(ii) The recurrence (𝑛+1)(𝑛+2)(𝑛+3)𝑈_𝑛+3=(3𝑛²+19𝑛+29)(𝑛+1)𝑈_𝑛+2−(3𝑛³+6𝑛²−7𝑛−13)𝑈_𝑛+1+(𝑛+2)³𝑈_𝑛, (2.6) provides two sequences of rational numbers (𝑝_𝑛)_𝑛≥0 and (𝑞_𝑛)_𝑛≥0 with 𝑝₀=−1, 𝑝₁=11, 𝑝₂=71 and 𝑞₀=0, 𝑞₁=8, 𝑞₂=56 such that (𝑝_𝑛/𝑞_𝑛)_𝑛≥0 converges to log(2)+𝛾.

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 convergence (see Theorems 2.4 and 6.2 below).

In the following we denote by 𝐵_2𝑛 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 𝐵_𝑛 from Corollary 2.2.) In this paper we will prove the following result.

Theorem 2.4. Let 𝑎≥1, 𝑏≥1, 𝑑≥42 and 𝑚≥1 be positive integers, and 𝑆_𝑛∶=^𝑎𝑛−1_𝑗=11𝑗−^𝑏𝑛−1_𝑗=11𝑗+2^𝑛−1_𝑗=11𝑗−^𝑛2−1_𝑗=11𝑗−12𝑛²+^𝑑𝑚_𝑗=1𝐵_2𝑗(12𝑗𝑛^2𝑗(1𝑎^2𝑗−1𝑏^2𝑗1+1)−𝑛^4𝑗),(𝑛≥1). (2.7) Then, |^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑘𝑆(2𝑑−1)𝑚+𝑘−1𝑑𝑚𝑑𝑚_{𝑘+𝑑𝑚}𝑎−𝛾−log𝑏|<𝑐₄⋅((1−1/𝑑)^𝑑(𝑑−1)4^𝑑)^𝑚, (2.8) where 𝑐₄ is some positive constant depending only on 𝑑.

3. Proof of Theorem 2.4

Lemma 3.1. One has for positive integers 𝑑 and 𝑚𝑘𝑔(𝑘)∶=(2𝑑−1)𝑚+𝑘−1𝑑𝑚𝑑𝑚<16^𝑑𝑚(0≤𝑘≤𝑑𝑚). (3.1)

Proof. Applying the well known inequality (^𝑔_ℎ)≤2^𝑔, we get 𝑘(2𝑑−1)𝑚+𝑘−1𝑑𝑚𝑑𝑚≤2^{(2𝑑−1)𝑚+𝑑𝑚−1}2^𝑑𝑚=2^{4𝑑𝑚−𝑚−1}<16^𝑑𝑚. (3.2) This proves the lemma.

𝑔(𝑘) takes its maximum value for 𝑘=𝑘₀ with 𝑘₀=5𝑑²−4𝑑+1−𝑑+12𝑚+𝒪(1), (3.3) which leads to a better bound than 16^𝑑𝑚 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 ^𝑛_𝑖=1𝑓(𝑖)=^𝑛₁𝑓(𝑥)𝑑𝑥+𝑓(1)+𝑓(𝑛)2+^𝑟_𝑗=1𝐵_2𝑗(2𝑗)!(𝑓^(2𝑗−1)(𝑛)−𝑓^(2𝑗−1)(1))+𝑅_𝑟, (3.4) where 𝑟∈ℕ is a suitable chosen parameter, and the remainder 𝑅_𝑟 is defined by a periodic Bernoulli polynomial 𝑃_2𝑟+1(𝑥), namely 𝑅_𝑟=1(2𝑟+1)!^𝑛₁𝑃_2𝑟+1(𝑥)𝑓^(2𝑟+1)(𝑥)𝑑𝑥, (3.5) with 𝑃_2𝑟+1(𝑥)=(−1)^𝑟−1(2𝑟+1)!^∞_𝑗=12sin(2𝜋𝑗𝑥)(2𝜋𝑗)^2𝑟+1. (3.6) Applying the summation formula to the function 𝑓(𝑥)=1/𝑥, we get (see [8, equation ( 5)] ) ^𝑛−1_𝑖=11𝑖1=log𝑛+2−1+2𝑛^𝑟_𝑗=1𝐵_2𝑗12𝑗(1−𝑛^2𝑗)−^𝑛₁𝑃_2𝑟+1(𝑥)𝑥^2𝑟+2𝑑𝑥,(𝑛,𝑟∈ℕ). (3.7) It follows that ^𝑛2−1_𝑖=𝑛1𝑖1−log𝑛=−12𝑛2𝑛²+^𝑟_𝑗=1𝐵_2𝑗(12𝑗𝑛^2𝑗−1𝑛^4𝑗)−^𝑛2_𝑛𝑃_2𝑟+1(𝑥)𝑥^2𝑟+2𝑑𝑥,(𝑛,𝑟∈ℕ). (3.8) 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 ^𝑎𝑛−1_{𝑖=𝑏𝑛}1𝑖𝑎−log𝑏1=(𝑏−1𝑎)1+2𝑛^𝑟_𝑗=1𝐵_2𝑗2𝑗𝑛^2𝑗(1𝑏^2𝑗−1𝑎^2𝑗)−^𝑎𝑛_𝑏𝑛𝑃_2𝑟+1(𝑥)𝑥^2𝑟+2𝑑𝑥,(𝑛,𝑟∈ℕ). (3.9) First, we estimate the integral on the right-hand side of (3.8). We have |^𝑛2_𝑛𝑃_2𝑟+1(𝑥)𝑥^2𝑟+2𝑑𝑥|≤^𝑛2_𝑛|𝑃_2𝑟+1(𝑥)|𝑥^2𝑟+2𝑑𝑥≤^∞_𝑛|𝑃_2𝑟+1(𝑥)|𝑥^2𝑟+2𝑑𝑥≤2(2𝑟+1)!^∞_𝑛1𝑥^∞2𝑟+2_𝑗=11(2𝜋𝑗)^2𝑟+1=𝑑𝑥2(2𝑟+1)!(2𝜋)^2𝑟+11[−(2𝑟+1)𝑥^2𝑟+1]^∞∞_{𝑥=𝑛𝑗=1}1𝑗^2𝑟+1=2(2𝑟)!(2𝜋)^2𝑟+1𝑛^2𝑟+1𝜁(2𝑟+1)<3(2𝑟)!(2𝜋)^2𝑟+1𝑛^2𝑟+1, (3.10) 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 |^𝑎𝑛_𝑏𝑛𝑃_2𝑟+1(𝑥)𝑥^2𝑟+2𝑑𝑥|≤^𝑎𝑛_𝑏𝑛|𝑃_2𝑟+1(𝑥)|𝑥^2𝑟+2≤𝑑𝑥_𝑛2𝑛|𝑃_2𝑟+1(𝑥)|𝑥^2𝑟+2𝑑𝑥≤3(2𝑟)!(2𝜋)^2𝑟+1𝑛^2𝑟+1. (3.11) 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 √2𝜋𝑚(𝑚𝑒)^𝑚<𝑚!<𝑚2𝜋(𝑚+1)(𝑒)^𝑚,(𝑚>0). (3.12) Then, it follows that |^{(𝑑𝑚+𝑘)2}_{𝑑𝑚+𝑘}𝑃_2𝑟+1(𝑥)𝑥^2𝑟+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≤3√3𝜋𝑑𝑚(2𝜋𝑑𝑚)(𝜋𝑒)^2𝑑𝑚, (3.13) and similarly we have |^{𝑎(𝑑𝑚+𝑘)}_{𝑏(𝑑𝑚+𝑘)}𝑃_2𝑟+1(𝑥)𝑥^2𝑟+23√𝑑𝑥|≤3𝜋𝑑𝑚(2𝜋𝑑𝑚)(𝜋𝑒)^2𝑑𝑚,(𝑑𝑚≥𝑎). (3.14) By using the definition of 𝑆_𝑛 in Theorem 2.4, the formula (1.1) for 𝑠_𝑛, and the identities (3.8), (3.9), it follows that 𝑆_𝑛𝑎−𝛾−log𝑏=(𝑠_𝑛−𝛾)+(𝑠_𝑛−𝑠²_𝑛)+(𝑠_𝑎𝑛−𝑠_𝑏𝑛1)−2𝑛²+^𝑟_𝑗=1𝐵_2𝑗(12𝑗𝑛^2𝑗(1𝑎^2𝑗−1𝑏^2𝑗1+1)−𝑛^4𝑗)=(𝑠_𝑛1−𝛾)+(12𝑛𝑏−1𝑎−1)+^𝑛2_𝑛𝑃_2𝑟+1(𝑥)𝑥^2𝑟+2𝑑𝑥−^𝑎𝑛_𝑏𝑛𝑃_2𝑟+1(𝑥)𝑥^2𝑟+2𝑑𝑥, (3.15) where 𝑟 is specified to 𝑟=𝑑𝑚 and 𝑛 to 𝑛=𝑑𝑚+𝑘. Moreover, we know from [4, Lemma  2] that ^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)=1,(𝑚≥1). (3.16) By setting 𝑛=𝑑𝑚+𝑘, the above formula for the series transformation of 𝑆_{𝑑𝑚+𝑘} simplifies to |^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)𝑆_{𝑑𝑚+𝑘}𝑎−𝛾−log𝑏|=|^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)(𝑠_{𝑑𝑚+𝑘}1−𝛾)+2(1𝑏−1𝑎−1)^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)+𝑑𝑚+𝑘_{𝑑𝑚𝑘=0}(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)^{(𝑑𝑚+𝑘)2}_{𝑑𝑚+𝑘}𝑃_2𝑟+1(𝑥)𝑥^2𝑟+2−𝑑𝑥_{𝑑𝑚𝑘=0}(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)^{𝑎(𝑑𝑚+𝑘)}_{𝑏(𝑑𝑚+𝑘)}𝑃_2𝑟+1(𝑥)𝑥^2𝑟+2𝑑𝑥|≤|^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)(𝑠_{𝑑𝑚+𝑘}−𝛾)|+^𝑑𝑚_𝑘=0𝑔(𝑘)|^{(𝑑𝑚+𝑘)2}_{𝑑𝑚+𝑘}𝑃_2𝑟+1(𝑥)𝑥^2𝑟+2+𝑑𝑥|_{𝑑𝑚𝑘=0}𝑔(𝑘)|^{𝑎(𝑑𝑚+𝑘)}_{𝑏(𝑑𝑚+𝑘)}𝑃_2𝑟+1(𝑥)𝑥^2𝑟+2𝑑𝑥|<𝐶_𝑑⋅((1−1/𝑑)^𝑑(𝑑−1)4^𝑑)^𝑚−2+^𝑑𝑚_𝑘=03√𝑔(𝑘)3𝜋𝑑𝑚𝜋𝑑𝑚(𝜋𝑒)^2𝑑𝑚, (3.17) 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 ^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)𝑑𝑚+𝑘 (3.18) vanishes, since for every real number 𝑥>−𝑑𝑚 we have ^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}(^{(2𝑑−1)𝑚+𝑘−1}_𝑑𝑚)(_𝑘^𝑑𝑚)=𝑑𝑚+𝑘+𝑥(1−(𝑑−1)𝑚+𝑥)⋯(𝑚+𝑥)(𝑑𝑚+𝑥)_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)^𝑑2(𝑑−1) (3.20) 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: |^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)𝑆_{𝑑𝑚+𝑘}𝑎−𝛾−log𝑏|<𝐶_𝑑⋅((1−1/𝑑)^𝑑(𝑑−1)4^𝑑)^𝑚−2+^𝑑𝑚_𝑘=03√3𝜋𝑑𝑚𝜋𝑑𝑚16^𝑑𝑚(𝜋𝑒)^2𝑑𝑚=𝐶_𝑑⋅((1−1/𝑑)^𝑑(𝑑−1)4^𝑑)^𝑚−2+3(𝑑𝑚+1)√3𝑑𝑚𝑑𝑚√𝜋14^𝑑𝑚(64(𝜋𝑒)^2𝑑𝑚_(3.2)₀₎<𝐶_𝑑⋅((1−1/𝑑)^𝑑(𝑑−1)4^𝑑)^𝑚−2+3(𝑑𝑚+1)√3𝑑𝑚𝑑𝑚2^𝑚√𝜋((1−1/𝑑)^𝑑(𝑑−1)4^𝑑)^𝑚≤𝐶_𝑑⋅((1−1/𝑑)^𝑑(𝑑−1)4^𝑑)^𝑚−2+85283𝑑𝜋((1−1/𝑑)^𝑑(𝑑−1)4^𝑑)^𝑚≤𝑐₄((1−1/𝑑)^𝑑(𝑑−1)4^𝑑)^𝑚. (3.21) 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 of 𝐒_𝐧

In this section we will investigate the size of the denominators 𝑏_𝑚 of our series transformations 𝑎_𝑚𝑏_𝑚=^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)𝑆_{𝑘+𝑑𝑚}, (4.1) for 𝑚 tending to infinity, where 𝑎_𝑚∈ℤ and 𝑏_𝑚∈ℕ are coprime integers.

Theorem 4.1. For every 𝑚≥1 there is an integer 𝑍_𝑚 with 𝑍_𝑚>0, 𝑏_𝑚|𝑍_𝑚, and log𝑍_𝑚∼12𝑑²𝑚²,(𝑚→∞). (4.2)

Proof. We will need some basic facts on the arithmetical functions 𝜗(𝑥) and 𝜓(𝑥). Let 𝜗(𝑥)=_𝑝≤𝑥log𝑝,(𝑥>1),𝜓(𝑥)=_𝑝≤𝑥[log𝑥log𝑝]log𝑝,(𝑥>1), (4.3) where 𝑝 is restricted on primes. Moreover, let 𝐷_𝑛∶=l.c.m(1,2,…,𝑛) for positive integers 𝑛. Then, 𝜓(𝑛)=log𝐷_𝑛,(𝑛≥1), (4.4)𝜓(𝑥)∼𝜗(𝑥)∼𝜋(𝑥)log𝑥∼𝑥,(𝑥→∞), (4.5) 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 𝐵_2𝑘: The denominators of 𝐵_2𝑘 are squarefree, and they are divisible exactly by those primes 𝑝 with (𝑝−1)∣2𝑘. Hence, 𝐵_{2𝑘𝑝≤2𝑘+1}𝑝∈ℤ,(𝑘=1,2,…). (4.6) Next, let max{𝑎,𝑏}≤𝑑𝑚≤𝑛≤2𝑑𝑚 (𝑛=𝑘+𝑑𝑚 are the subscripts of 𝑆_{𝑘+𝑑𝑚} in Theorem 2.4). First, we consider the following terms from the series transformation in 𝑆_𝑚: ^𝑎𝑛−1_𝑗=11𝑗−^𝑏𝑛−1_𝑗=11𝑗+2^𝑛−1_𝑗=11𝑗−^𝑛2−1_𝑗=11𝑗=∶^𝑛2−1_𝑗=1𝑒_𝑗𝑗, (4.7) with 𝑒_𝑗∶=1,if1≤𝑗≤𝑛−1−1,if𝑛≤𝑗≤𝑏𝑛−10,if𝑏𝑛≤𝑗≤𝑎𝑛−1−1,if𝑎𝑛≤𝑗≤𝑛²−1(𝑎≥𝑏),𝑒_𝑗∶=1,if1≤𝑗≤𝑛−1−1,if𝑛≤𝑗≤𝑎𝑛−1−2,if𝑎𝑛≤𝑗≤𝑏𝑛−1−1,if𝑏𝑛≤𝑗≤𝑛²−1.(𝑎<𝑏). (4.8) For every 𝑚≥1 there is a rational 𝑥_𝑚/𝑦_𝑚 defined by 𝑥_𝑚𝑦_𝑚=^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)^{(𝑘+𝑑𝑚)2−1}_𝑗=1𝑒_𝑗𝑗, (4.9) where 𝑥_𝑚∈ℤ, 𝑦_𝑚∈ℕ, (𝑥_𝑚,𝑦_𝑚)=1, and 𝑦_𝑚∣𝑌_𝑚∶=𝐷_4𝑑²_𝑚2,(𝑑𝑚≥max{𝑎,𝑏}). (4.10) Similarly, we define rationals 𝑢_𝑚/𝑣_𝑚 by𝑢_𝑚𝑣_𝑚=^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}1𝑔(𝑘)×(−2(𝑘+𝑑𝑚)²+^𝑑𝑚_𝑗=1𝐵_2𝑗(12𝑗(𝑘+𝑑𝑚)^2𝑗(1𝑎^2𝑗−1𝑏^2𝑗1+1)−(𝑘+𝑑𝑚)^4𝑗)), (4.11) where 𝑢_𝑚∈ℤ, 𝑣_𝑚∈ℕ and (𝑢_𝑚,𝑣_𝑚)=1. We have (𝑘+𝑑𝑚)^2𝑗∣(𝑘+𝑑𝑚)^4𝑑𝑚,(0≤𝑘≤𝑑𝑚,1≤𝑗≤𝑑𝑚). (4.12) Therefore, using the conclusion (4.6) from von Staudt's theorem, we get 𝑣_𝑚∣𝑉_𝑚∶=2(𝑎𝑏)^2𝑑𝑚𝐷_𝑑𝑚(_{𝑝≤2𝑑𝑚+1}𝑝)(𝐷_2𝑑𝑚)^4𝑑𝑚,(𝑑𝑚≥max{𝑎,𝑏}). (4.13) Note that 𝐷_2𝑑𝑚=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 𝑏_𝑚∣𝑍_𝑚∶=2(𝑎𝑏)^2𝑑𝑚𝐷_𝑑𝑚𝐷_4𝑑²_𝑚2(𝐷_2𝑑𝑚)^4𝑑𝑚(_{𝑝≤2𝑑𝑚+1}𝑝). (4.14) Hence we have from (4.4) and (4.5) that log𝑍_𝑚=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→∞). (4.15) The theorem is proved.

Remark 4.2. On the one side we have shown that log𝑌_𝑚∼4𝑑²𝑚² and log𝑉_𝑚∼8𝑑²𝑚². On the other side, every prime 𝑝 dividing 𝑉_𝑚 satisfies 𝑝≤max{𝑎,𝑏,𝑑𝑚,2𝑑𝑚+1,2𝑑𝑚}=2𝑑𝑚+1 and therefore 𝑝 divides 𝑌_𝑚=𝐷_4𝑑²_𝑚2. Conversely, all primes 𝑝 with 2𝑑𝑚+1<𝑝<4𝑑²𝑚² divide 𝑌_𝑚, but not 𝑉_𝑚. That means: 𝑉_𝑚 is much bigger than 𝑌_𝑚, but 𝑉_𝑚 is formed by powers of small primes, whereas 𝑌_𝑚is divisible by many big primes.

5. Simplification of the Transformed Series

Let 𝑅_𝑛1∶=−2𝑛²+^𝑑𝑚_𝑗=1𝐵_2𝑗(12𝑗𝑛^2𝑗(1𝑎^2𝑗−1𝑏^2𝑗1+1)−𝑛^4𝑗), (5.1) such that 𝑆_𝑛=^𝑎𝑛−1_𝑗=11𝑗−^𝑏𝑛−1_𝑗=11𝑗+2^𝑛−1_𝑗=11𝑗−^𝑛2−1_𝑗=11𝑗+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 𝑆_𝑛−𝑅_𝑛. Therefore we have to estimate the contribution of 𝑅_{𝑘+𝑑𝑚} to the series transformation in Theorem 2.4. For this purpose, we define 𝐸_𝑚∶=^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑘𝑅(2𝑑−1)𝑚+𝑘−1𝑑𝑚𝑑𝑚_{𝑘+𝑑𝑚}1=−2^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)(𝑑𝑚+𝑘)²+^𝑑𝑚_𝑗=1𝐵_2𝑗12𝑗((𝑎^2𝑗−1𝑏^2𝑗+1)^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)(𝑑𝑚+𝑘)^2𝑗−^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)(𝑑𝑚+𝑘)^4𝑗). (5.3) A major step in estimating 𝐸_𝑚 is to expressthe sums on the right-hand side by integrals.

Lemma 5.1. For positive integers 𝑑,𝑗 and 𝑚 one has ^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)(𝑑𝑚+𝑘)^2𝑗=−(−1)^𝑑𝑚(𝑑𝑚)!(2𝑗−1)!¹₀𝑢^𝑚(log𝑢)^2𝑗−1𝜕^𝑑𝑚𝜕𝑢^𝑑𝑚(𝑢^{(2𝑑−1)𝑚−1}(1−𝑢)^𝑑𝑚)𝑑𝑢. (5.4)

Proof. For integers 𝑘,𝑟 and a real number 𝜌 with 𝑘+𝜌>0 the identity 1(𝑘+𝜌)^𝑟=1(𝑟−1)!^∞₀𝑒^{−(𝑘+𝜌)𝑡}𝑡^𝑟−1𝑑𝑡 (5.5) 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 ^2𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)(𝑘+𝑑𝑚)^2𝑗=−(−1)^𝑑𝑚(2𝑗−1)!^𝑑𝑚_𝑘=0(−1)^𝑘𝑔(𝑘)¹₀𝑢^{𝑘+𝑑𝑚−1}(log𝑢)^2𝑗−1𝑑𝑢=−(−1)^𝑑𝑚(2𝑗−1)!¹₀(^𝑑𝑚_𝑘=0(−1)^𝑘𝑔(𝑘)𝑢^{𝑘+(𝑑−1)𝑚−1})𝑢^𝑚(log𝑢)^2𝑗−1𝑑𝑢. (5.6) The sum inside the brackets of the integrand can be expressed by using the equation ^𝑛_𝑘=0(−1)^𝑘𝑛𝑛𝑘𝑢𝑛+𝜏+𝑘_𝜏+𝑘=𝜕^𝑛𝜕𝑢^𝑛(𝑢^𝑛+𝜏(1−𝑢)^𝑛𝑛!),(𝑛,𝜏∈ℕ∪{0}), (5.7) 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 case 𝑗=1, in which we express the finite sum by a double integral on a rational function.

Corollary. For every positive integer 𝑚 one has ^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)(𝑑𝑚+𝑘)²=(−1)^{(𝑑−1)𝑚1}₀¹₀(1−𝑢)^𝑑𝑚(1−𝑤)^𝑚𝑢^{(2𝑑−1)𝑚−1}𝑤^{(𝑑−1)𝑚−1}(1−(1−𝑢)𝑤)^𝑑𝑚+1𝑑𝑢𝑑𝑤. (5.8)

Proof. Set 𝑗=1 in Lemma 5.1, and note that log𝑢=−(1−𝑢)¹₀𝑑𝑤.1−(1−𝑢)𝑤 (5.9) Hence, ^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)(𝑑𝑚+𝑘)²=−(−1)^𝑑𝑚(𝑑𝑚)!¹₀𝑢^𝑚𝜕log𝑢^𝑑𝑚𝜕𝑢^𝑑𝑚(𝑢^{(2𝑑−1)𝑚−1}(1−𝑢)^𝑑𝑚=)𝑑𝑢(−1)^𝑑𝑚(𝑑𝑚)!¹₀¹₀(1−𝑢)𝑢^𝑚𝜕1−(1−𝑢)𝑤𝜕𝑢^{𝑑𝑚𝑑𝑚}(𝑢^{(2𝑑−1)𝑚−1}(1−𝑢)^𝑑𝑚)𝑑𝑢𝑑𝑤. (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: 𝑢^𝑠=1−(1−𝑢)𝑤^𝑠−1_𝜈=0(𝑤−1)^𝜈𝑤^𝜈+1𝑢^{𝑠−𝜈−1}+(𝑤−1𝑤)^𝑠1.1−(1−𝑢)𝑤 (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𝜕𝑢^𝑑𝑚(𝑢^𝑠1−(1−𝑢)𝑤)=(𝑤−1𝑤)^𝑠(−1)^𝑑𝑚(𝑑𝑚)!𝑤^𝑑𝑚(1−(1−𝑢)𝑤)^𝑑𝑚+1. (5.12) Therefore, we get from (5.10) by iterated integrations by parts: ^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)(𝑑𝑚+𝑘)²=1(𝑑𝑚)!¹₀¹₀𝑢^{(2𝑑−1)𝑚−1}(1−𝑢)^𝑑𝑚𝜕^𝑑𝑚𝜕𝑢^𝑑𝑚(𝑢^𝑚−𝑢^𝑚+1=11−(1−𝑢)𝑤)𝑑𝑢𝑑𝑤(𝑑𝑚)!¹₀¹₀𝑢^{(2𝑑−1)𝑚−1}(1−𝑢)^𝑑𝑚((𝑤−1𝑤)^𝑚−(𝑤−1𝑤)^𝑚+1)×(−1)^𝑑𝑚(𝑑𝑚)!𝑤^𝑑𝑚𝑑𝑢𝑑𝑤(1−(1−𝑢)𝑤)^𝑑𝑚+1. (5.13) The corollary is proved by noting that (𝑤−1𝑤)^𝑚−(𝑤−1𝑤)^𝑚+1=(−1)^𝑚(1−𝑤)^𝑚𝑤^𝑚+1. (5.14)

6. Estimating 𝐄_𝐦

In this section we estimate 𝐸_𝑚 defined in (5.3). Substituting 1−𝑢 for 𝑢 into the integral in Lemma 5.1 and applying iterated integration by parts, we get ^𝑑𝑚_𝑘=0(−1)^{𝑑𝑚+𝑘}𝑔(𝑘)(𝑑𝑚+𝑘)^2𝑗=−(−1)^𝑑𝑚(𝑑𝑚)!(2𝑗−1)!¹₀𝜕^𝑑𝑚𝜕𝑢^𝑑𝑚((1−𝑢)^𝑚(log(1−𝑢))^2𝑗−1)((1−𝑢)^{(2𝑑−1)𝑚−1}𝑢^𝑑𝑚)𝑑𝑢. (6.1) Set 𝑓(𝑢)∶=(1−𝑢)^𝑚(log(1−𝑢))^2𝑗−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 𝑓^(𝑘)(𝑎)=𝑘!2𝜋𝑖_𝐶𝑓(𝑧)(𝑧−𝑎)^𝑘+1𝑑𝑧 (6.3) 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 𝑓^(𝑘)(0)=𝑘!2𝜋𝑅^{𝑘𝜋−𝜋}𝑒^{−𝑖𝑘𝜙}(1−𝑅𝑒^𝑖𝜙)^𝑚log^2𝑗−1(1−𝑅𝑒^𝑖𝜙)𝑑𝜙. (6.4) 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−𝑅𝑒^𝑖𝜙=1−𝑅cos𝜙−𝑖𝑅sin𝜙, we get |1−𝑅𝑒^𝑖𝜙|=1+𝑅²−2𝑅cos𝜙=∶𝐴(𝑅,𝜙),arg(1−𝑅𝑒^𝑖𝜙)=−arctan(𝑅sin𝜙1−𝑅cos𝜙). (6.5) Hence, log(1−𝑅𝑒^𝑖𝜙)=ln1+𝑅²−2𝑅cos𝜙−𝑖arctan(𝑅sin𝜙)=11−𝑅cos𝜙2ln(𝐴(𝑅,𝜙))−𝑖arctan(𝑅sin𝜙1−𝑅cos𝜙). (6.6) Thus, it follows from (6.4) that

|𝑓^(𝑘)(0)|≤𝑘!2𝜋𝑅^{𝑘𝜋−𝜋}|1−𝑅𝑒^𝑖𝜙|^𝑚⋅|log(1−𝑅𝑒^𝑖𝜙)|^2𝑗−1=𝑑𝜙𝑘!2𝜋𝑅^{𝑘𝜋−𝜋}(𝐴(𝑅,𝜙))^𝑚/2(14ln²(𝐴(𝑅,𝜙))+arctan²(𝑅sin𝜙1−𝑅cos𝜙))^{(2𝑗−1)/2}=𝑑𝜙𝑘!𝜋𝑅^𝑘𝜋0(𝐴(𝑅,𝜙))^𝑚/2(14ln²(𝐴(𝑅,𝜙))+arctan²(𝑅sin𝜙1−𝑅cos𝜙))^{(2𝑗−1)/2}𝑑𝜙. (6.7) From 0<𝑅<1 we conclude on 0<(1−𝑅)²=1+𝑅²−2𝑅≤1+𝑅²−2𝑅cos𝜙=𝐴(𝑅,𝜙)<4,(0≤𝜙≤𝜋),0≤𝑅sin𝜙≤1−𝑅cos𝜙sin𝜙1−cos𝜙,(0<𝜙≤𝜋). (6.8) Since arctan is a strictly increasing function, we get arctan(𝑅sin𝜙1−𝑅cos𝜙)≤arctan(sin𝜙1−cos𝜙)=arctancot(𝜙2)=arctan(tan(𝜋−𝜙2=))𝜋−𝜙2,(0<𝜙≤𝜋). (6.9) For 0<𝑅<1, this upper bound also holds for 𝜙=0. Finally, we note that 𝑅^𝑘=(1−1/2𝑘)^𝑘≥1/2. Altogether, we conclude from (6.7) on |𝑓^(𝑘)(0)|≤𝑘!𝜋𝑅^𝑘𝜋04^𝑚/2(ln²44+arctan²(sin𝜙1−cos𝜙))^{(2𝑗−1)/2}≤2𝑑𝜙_𝑚+1𝑘!𝜋^𝜋₀(ln²