DOCSLIB.ORG

Some Integrals Involving the Stieltjes Constants: Part II

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Some Integrals Involving the Stieltjes Constants: Part II Some integrals involving the Stieltjes constants: Part II Donal F. Connon [email protected] 11 April 2011 Abstract Some new integrals involving the Stieltjes constants are developed in this paper. CONTENTS Page 1. Introduction 1 2. A useful formula for the Stieltjes constants 2 3. An application of the Abel-Plana summation formula 6 4. A family of integral representations of the Stieltjes constants 16 5. An application of the alternating Hurwitz zeta function 35 1. Introduction The Stieltjes constants γ p ()x are the coefficients of the Laurent expansion of the Hurwitz zeta function ς (,sx ) about s =1 ∞∞p 11(1)− p (1.1) (,sx ) ()(x s 1) ςγ==+∑∑s p − np==00()nx+− s 1 p ! where γ p ()x are known as the generalised Stieltjes constants and we have [41] (1.2) γ 0 ()x =−ψ ()x where ψ ()x is the digamma function. With x =1 equation (1.1) reduces to the Riemann zeta function ∞ p 1(1)− p ςγ()ss=+∑ p ( − 1) sp−1!p=0 As previously noted in [20], using (1.1) it is easily seen that the difference of two Stieltjes constants may be represented by p p ∂ (1.3) γγpp()x −=− ()ys (1)lim [(,)ςx − ς (,sy )] s→1 ∂s p 2. A useful formula for the Stieltjes constants We recall Hasse’s formula [31] for the Hurwitz zeta function which is valid for all s ∈C except s =1 (in this form, it is valid in the limit as s →1) ∞ 1(j ⎛⎞j −1)k (2.1) (1)(,)ssx −=ς ∑∑⎜⎟ s−1 jk==00jx++1()⎝⎠k k and differentiation with respect to x gives us ∂−∞ 1(j ⎛⎞j 1)k (,sx ) ς =−∑∑⎜⎟ s ∂+x jk==00jx1(⎝⎠k +k) and we then have nk∞ j n ∂∂ n+1 1(1)log(⎛⎞j −x +k) (2.2) (,sx ) (1) nsς =− ∑∑⎜⎟ ∂∂sx jk==00j+1()⎝⎠k xk+ We note that the partial derivatives commute in the region where ς (,)sx is analytic and hence we have ∂∂nn ∂∂ ςς(,sx )= (, sx ) ∂∂sxnn ∂∂ xs Evaluation of (2.2) at s = 0 results in ∞ j ∂ ()nn+1 1 ⎛⎞j kn (2.3) ς (0,x )=− ( 1) ∑∑⎜⎟(− 1) log (xk + ) ∂+xjjk==001 ⎝⎠k We may write (1.1) as ∞ n 1(1)− n (2.4) ςγ(1,)sx+=+∑ n ()xs snn=0 ! and we have the Maclaurin expansion 2 1 ∞ (1)− n+1 R ()x (2.5) ς (,sx )=+∑ n sn sn−1!n=0 where n n+1 ∂ Rxn ()=− (1)n ς (,)sx ∂s s=0 R()xRx= 2 () is referred to as the Deninger R − function, after Deninger [28] who introduced it in 1984. Differentiating (2.5) with respect to x gives us ∂ ∞ (1)− n+1 R′ ()x (2.6) ς (,sx )= ∑ n sn ∂xnn=0 ! We note that ∂ ςς(,sx )=− s ( s + 1,) x ∂x and, using (2.4), this is equal to ∞ n+1 (1)− n+1 (2.7) =−1( +∑ γ n x)s n=0 n! and comparing the coefficients of (2.6) and (2.7) Chakraborty, Kanemitsu and Kuzumaki [12] deduced the important identity ∂∂n ′ n+1 (2.8) Rnn()xs=− (1)n ςγ (,)x=−n−1 ()x ∂∂sx s=0 This may be compared with the more familiar formula for the Stieltjes constants where the limit is evaluated at s =1 ∂n−1 ⎡⎤1 (2.9) (1)−−=n+1 ςγ (,)sx −()x n−1 ⎢⎥n−1 ∂−ss⎣⎦1 s=1 We see from (2.3) that (2.8) is equal to ∞ j 1 ⎛⎞j kn =−∑∑⎜⎟(1)log(x +k ) jk==00j +1 ⎝⎠k 3 and hence we easily deduce that ∞ j 11⎛⎞j kn+1 (2.10) γ n ()x =− ∑∑⎜⎟(1)log(−xk + ) nj++11jk==00⎝⎠k which was previously obtained in 2007 in [20]. In [20, Eq. (4.3.231)] we also noted that Rnn′()xnx= − γ −1 () in the equivalent form for x > 0 x (1)− n+1 γςς()tdt=−⎡ (1)nn++(0,x )(1) (0)⎤ ∫ n ⎣ ⎦ 1 n +1 but the usefulness of this simple identity was not then fully appreciated by the author. The formula (2.8) features throughout the rest of this paper where it is used to simplify the derivation of some known identities and also to produce some new ones. Remark (i) The equivalent formula to (2.5) as reported by Chakraborty et al. [12] did not include the 1 term ; it seems to me that it should be so included if only to concur with the analysis s −1 previously carried out by Sitaramachandrarao [37] in 1986 where he considered the Maclaurin series for the Riemann zeta function 1 ∞ (1)− n δ (2.11) ς ()ss+=∑ n n 1!− snn=0 where m m ⎡⎤nn1 n (2.12) δn =−−lim logkxdx log log m m→∞ ⎢⎥∑ ∫ ⎣⎦k =1 1 2 nn() =−(1)⎣⎦⎡⎤ς (0) +n ! 1 This additional term of course vanishes when equation (2.5) is differentiated with s −1 respect to x and thus (2.8) continues to remain valid. 4 Remark (ii) We note from (2.8) that with n =1 ∂∂ R10′()xsxx==ςγ (,)− ()=ψ ()x ∂∂sx s=0 or equivalently ∂ ςψ′(0,x )= (x ) ∂x and integration then results in (2.13) ς ′′(0,tt )−=Γς (0) log ( ) We have Legendre’s duplication formula for the gamma function [38, p.7] ⎛⎞11 logΓ=Γ+Γ++− (2tt ) log ( ) log⎜⎟ tt (2 1)log 2 − logπ ⎝⎠22 and substituting (2.13) gives us ⎛⎞11 ς ′′′(0, 2tt )=++−+−−ςς (0, )⎜⎟ 0, t ς ′ (0) (2 t 1) log 2 logπ ⎝⎠22 and with t =1/2 we obtain ⎛⎞11 ς ′′⎜⎟0,=+ςπ (0) log ⎝⎠22 We recall the identity [30] ⎛⎞1 s ς ⎜⎟ss,[21](=−ς ) ⎝⎠2 and differentiation results in ⎛⎞1 ss ςςς′′⎜⎟sss,[21]()()2log=− + 2 ⎝⎠2 so that ⎛⎞11 ςς′⎜⎟0,==− (0)log 2 log 2 ⎝⎠22 5 This then gives us the well-known result 1 (2.14) ς ′(0)=− log(2π ) 2 and hence we have obtained Lerch’s identity [7] in a very direct manner without the need to resort to the functional equation for the Riemann zeta function 1 (2.15) ς ′(0,tt )=Γ− log ( ) log(2π ) 2 Remark (iii) It should be noted that the Stieltjes constant γ1 referred to in Eq. (2.15) of Deninger’s paper [28] should be increased by a factor of 2 (this difference arises because Deninger [28, p.174] employed a different definition in the Laurent expansion of the Hurwitz zeta function; this value was also inconsistently employed in Eq. (1.26) of the paper by Chakraborty et al. [12]). 3. An application of the Abel-Plana summation formula Adamchik [2] has recently reported that the Hermite integral for the Hurwitz zeta function may be derived from the Abel-Plana summation formula [38, p.90] ∞ 1(∞∞fix)−− f( ix) (3.1) f ()kf=+ (0) fxdxi () + dx ∑ ∫∫2π x k=0 2100e − which applies to functions which are analytic in the right-hand plane and satisfy the convergence condition limefxiy−2π y (+ )= 0 uniformly on any finite interval of x . y→∞ Derivations of the Abel-Plana summation formula are given in [39, p.145] and [40, p.108]. The Hermite integral for the Hurwitz zeta function may be derived as follows. Letting f ()kku=+ ( )−s we obtain ∞ 1(uu−−ss1 ∞ uixuix+−−)−s() − s (3.2) ς (,su )==++i dx ∑ sx∫ 2π k =0 ()2ku+− s 10 e − 1 Then, noting that (u+−−= ix)()()()−−−−s u ixsisi reθθ − re −s 6 =−re−−sisis[]θ eθ 2 (3.2.1) ()()uix+−−=−−ss uix sin(tan(/s−1 xu)) iu()22/2+ x s we may write (3.2) as Hermite’s integral for the Hurwitz zeta function ς (,)su uu−−ss11∞ sin( s tan− ( xu / )) (3.3) ς (,su )=+ +2 dx ∫ 22/22sxπ 21()(1)suxe−+0 − Differentiating (3.2) with respect to u ∂+su−−ss11∞ ()()u ix−− − u− ix −−s1 (3.4) ς (,su )=− −u−s − is dx ∫ 2π x ∂−ue210 We note that if f ()()ssgs= then the Leibniz differentiation formula results in f (1)nn++()ssgs=++ (1) () ( n 1) gs()n () so that (3.5) fng(1)nn+ (0)=+ ( 1)() (0) and hence we obtain n+1 ∂∂n +1 nn n++11 n n+1 ς (,su )=− (1)log −u − (1) − log u ∂∂sus=0 2 u ∞ ()log()()log()uixuixuixuix−+−+−nn −−in(1)(n + 1) dx ∫ 222π x 0 ()(1)uxe+− and comparing this with (2.8) we readily obtain the integral formula originally obtained by Coffey [17] in 2007 for the Stieltjes constants 11∞ ()log()()log(uixuixuixuix−+−+−nn) (3.6) γ ()uu=− lognn log+1 ui + dx n ∫ 222π x 21un++0 ()uxe(1−) This derivation is slightly more direct than the one originally provided by Coffey [17] (and is also simpler than my previous proof in [22]). □ Chen [13] has recently shown that for u > 0 and s > 0 7 ∞ −1 2s−−uy2 21 s 2 in[sxutan(/)] (3.7) eysin( xydy ) = ∫ 22/2s Γ+()su0 ()x and hence we have using Hermite’s integral (3.3) −−ss1 ∞∞ uu 41 −−uy2 21 s 2 ς (,s u )=+ + dx e ysin( xy ) dy ∫∫2π x 21()1sse−Γ00 − −−ss12∞∞ uu 4s−−uy2 21 s in(xy) =+ + e y dy dx ∫∫2π x 21()ss−Γ00 e−1 Using Legendre’s relation [39, p.122] ∞ sin(xt ) 1 1 1 1t 1 (3.7.1) 2cdx =−+=−oth ∫ 2π xt 0 eet−−11222t this results in −−ss1 ∞ uu 211−−uy2 21 s ⎡ 1⎤ ς (,su )=+ + e y −+dy ∫ ⎢ y2 2 ⎥ 21()ss−Γ0 ⎣e −1 y2⎦ With the substitution vy= 2 this becomes −−ss1 ∞ uu 111−−uv s 1 ⎡ 1⎤ (3.8) ς (,su )=+ + e v −+dv ∫ ⎢ v ⎥ 21()ss−Γ0 ⎣ ev − 12⎦ which is reported in [38, p.92] as being valid for Re ()s > −1.
Load more

Recommended publications
  • A Note on Some Constants Related to the Zeta–Function and Their Relationship with the Gregory Coefficients
    A note on some constants related to the zeta–function and their relationship with the Gregory coefficients Iaroslav V. Blagouchine∗ Steklov Institute of Mathematics at St.-Petersburg, Russia. Marc–Antoine Coppo Université Côte d’Azur, CNRS, LJAD (UMR 7351), France. Abstract In this article, new series for the first and second Stieltjes constants (also known as generalized Eu- ler’s constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so–called Gregory coefficients, which are also known as (reciprocal) log- arithmic numbers, Cauchy numbers of the first kind and Bernoulli numbers of the second kind. In addition, two interesting series with rational terms for Euler’s constant g and the constant ln 2p are given, and yet another generalization of Euler’s constant is proposed and various formulas for the cal- culation of these constants are obtained. Finally, we mention in the paper that almost all the constants considered in this work admit simple representations via the Ramanujan summation. Keywords: Stieltjes constants, Generalized Euler’s constants, Series expansions, Ramanujan summation, Harmonic product of sequences, Gregory’s coefficients, Logarithmic numbers, Cauchy numbers, Bernoulli numbers of the second kind, Stirling numbers of the first kind, Harmonic numbers. I. Introduction and definitions The zeta-function ¥ z(s) := ∑ n−s , Re s > 1, n=1 is of fundamental and long-standing importance in analytic number theory, modern analysis, theory of L–functions, prime number theory and in a variety of other fields. The z–function is a meromorphic function on the entire complex plane, except at the point s = 1 at which it has one simple pole with residue 1.
    [Show full text]
  • Two Series Expansions for the Logarithm of the Gamma Function Involving Stirling Numbers and Containing Only Rational −1 Coefficients for Certain Arguments Related to Π
    J. Math. Anal. Appl. 442 (2016) 404–434 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational −1 coefficients for certain arguments related to π Iaroslav V. Blagouchine ∗ University of Toulon, France a r t i c l e i n f o a b s t r a c t Article history: In this paper, two new series for the logarithm of the Γ-function are presented and Received 10 September 2015 studied. Their polygamma analogs are also obtained and discussed. These series Available online 21 April 2016 involve the Stirling numbers of the first kind and have the property to contain only Submitted by S. Tikhonov − rational coefficients for certain arguments related to π 1. In particular, for any value of the form ln Γ( 1 n ± απ−1)andΨ ( 1 n ± απ−1), where Ψ stands for the Keywords: 2 k 2 k 1 Gamma function kth polygamma function, α is positive rational greater than 6 π, n is integer and k Polygamma functions is non-negative integer, these series have rational terms only. In the specified zones m −2 Stirling numbers of convergence, derived series converge uniformly at the same rate as (n ln n) , Factorial coefficients where m =1, 2, 3, ..., depending on the order of the polygamma function. Explicit Gregory’s coefficients expansions into the series with rational coefficients are given for the most attracting Cauchy numbers −1 −1 1 −1 −1 1 −1 values, such as ln Γ(π ), ln Γ(2π ), ln Γ( 2 + π ), Ψ(π ), Ψ( 2 + π )and −1 Ψk(π ).
    [Show full text]
  • Expansions of Generalized Euler's Constants Into the Series Of
    Journal of Number Theory 158 (2016) 365–396 Contents lists available at ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt Expansions of generalized Euler’s constants into −2 the series of polynomials in π and into the formal enveloping series with rational coefficients only Iaroslav V. Blagouchine 1 University of Toulon, France a r t i c l e i n f o a b s t r a c t Article history: In this work, two new series expansions for generalized Received 1 January 2015 Euler’s constants (Stieltjes constants) γm are obtained. The Received in revised form 26 June first expansion involves Stirling numbers of the first kind, 2015 − contains polynomials in π 2 with rational coefficients and Accepted 29 June 2015 converges slightly better than Euler’s series n−2. The Available online 18 August 2015 Communicated by David Goss second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and Keywords: involves Bernoulli numbers with a non-linear combination of Generalized Euler’s constants generalized harmonic numbers. It also permits to derive an Stieltjes constants interesting estimation for generalized Euler’s constants, which Stirling numbers is more accurate than several well-known estimations. Finally, Factorial coefficients in Appendix A, the reader will also find two simple integral Series expansion definitions for the Stirling numbers of the first kind, as well Divergent series Semi-convergent series an upper bound for them. Formal series © 2015 Elsevier Inc. All rights reserved. Enveloping series Asymptotic expansions Approximations Bernoulli numbers Harmonic numbers Rational coefficients Inverse pi E-mail address: [email protected].
    [Show full text]
  • Series Representation of the Riemann Zeta Function and Other Results: Complements to a Paper of Crandall
    MATHEMATICS OF COMPUTATION Volume 83, Number 287, May 2014, Pages 1383–1395 S 0025-5718(2013)02755-X Article electronically published on July 29, 2013 SERIES REPRESENTATION OF THE RIEMANN ZETA FUNCTION AND OTHER RESULTS: COMPLEMENTS TO A PAPER OF CRANDALL MARK W. COFFEY Abstract. We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation throughout the whole complex plane. Additionally, we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hur- witz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incomplete Gamma function. Finally, we evaluate certain integrals including ζ(s) η(s) Res=c s ds and Res=c s ds,withζ the Riemann zeta function and η its alternating form. 1. Introduction and statement of results Very recently R. Crandall has described series-based algorithms for computing multiprecision values of several important functions appearing in analytic number theory and the theory of special functions [9]. These functions include the Lerch ∞ n s transcendent Φ(z,s,a)= n=0 z /(n + a) , and many of its special cases, and Epstein zeta functions. In the following we let Bn(x)andEn(x) be the Bernoulli and Euler polynomials, respectively, Γ the Gamma function, and Γ(x, y) the incomplete Gamma function (e.g., [1, 2, 13]). An example series representation from [9, (27)] is that for the Hurwitz zeta function ζ(s, a), ∞ ∞ 1 Γ[s, λ(n + a)] 1 B (a) λm+s−1 (1.1) ζ(s, a)= + (−1)m m , Γ(s) (n + a)s Γ(s) m! (m + s − 1) n=0 m=0 with free parameter λ ∈ [0, 2π).
    [Show full text]
  • Arxiv:1401.3724V1 [Math.NT] 15 Jan 2014
    A theorem for the closed–form evaluation of the first generalized Stieltjes constant at rational arguments Iaroslav V. Blagouchine∗ University of Toulon, France. Abstract The Stieltjes constants, also known as generalized Euler’s constants, are of fundamental and long– standing importance in modern analysis, number theory, theory of special functions and other disci- plines. Recently, it was conjectured that the first generalized Stieltjes constant at rational argument γ1(k/n), where k and n are positive integers such that k < n, may be always expressed by means of the Euler’s constant γ, the first Stieltjes constant γ1 , the logarithm of the Γ–function at rational argu- ment(s) and some relatively simple, perhaps even elementary, function. This conjecture was based on the evaluation of γ1(1/2), γ1(1/3), γ1(2/3), γ1(1/4), γ1(3/4), γ1(1/6), γ1(5/6), which could be expressed in this way. In this short article we complete this previous study by deriving an elegant theorem which allows to evaluate the first generalized Stieltjes constant at any rational argument. Besides, several related summation formulæ involving the first generalized Stieltjes constant and the Digamma function are also derived and discussed. Key words: Stieltjes constants, generalized Euler’s constants, Special constants, Number theory, Zeta function, Gamma function, Rational arguments, logarithmic integrals, Malmsten’s integrals, logarithmic series, theory of functions of a complex variable, orthogonal expansions. I. Introduction and notations I.1. Introduction The ζ–functions are one of more important special functions in modern analysis and theory of functions. The most known and frequently encountered of ζ–functions are Riemann and Hurwitz ζ–functions.
    [Show full text]
  • Euler's Constant: Euler's Work and Modern Developments
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, Number 4, October 2013, Pages 527–628 S 0273-0979(2013)01423-X Article electronically published on July 19, 2013 EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS JEFFREY C. LAGARIAS Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ··· bearing his name, together with some of his related work on the gamma function, values of the zeta function, and diver- gent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant. Contents 1. Introduction 528 2. Euler’s work 530 2.1. Background 531 2.2. Harmonic series and Euler’s constant 531 2.3. The gamma function 537 2.4. Zeta values 540 2.5. Summing divergent series: the Euler–Gompertz constant 550 2.6. Euler–Mascheroni constant; Euler’s approach to research 555 3. Mathematical developments 557 3.1. Euler’s constant and the gamma function 557 3.2. Euler’s constant and the zeta function 561 3.3. Euler’s constant and prime numbers 564 3.4. Euler’s constant and arithmetic functions 565 3.5. Euler’s constant and sieve methods: the Dickman function 568 3.6. Euler’s constant and sieve methods: the Buchstab function 571 3.7.
    [Show full text]
  • Journal of Computational and Applied Mathematics 225 (2009) 338–346
    Journal of Computational and Applied Mathematics 225 (2009) 338–346 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam An efficient algorithm for the Hurwitz zeta and related functions Mark W. Coffey Department of Physics, Colorado School of Mines, Golden, CO 80401, United States article info a b s t r a c t Article history: A simple class of algorithms for the efficient computation of the Hurwitz zeta and related Received 16 June 2007 special functions is given. The algorithms also provide a means of computing fundamental mathematical constants to arbitrary precision. A number of extensions as well as numerical MSC: examples are briefly described. The algorithms are easy to implement and compete with 11M06 Euler–Maclaurin summation-based methods. 11M35 ' 2008 Elsevier B.V. All rights reserved. 33B15 Keywords: Hurwitz zeta function Riemann zeta function Polygamma function Lerch zeta function Algorithm Integral representation Generalized harmonic numbers Stieltjes constants 1. Introduction In this paper we present efficient algorithms for computing the Riemann and Hurwitz zeta functions in the complex plane. The algorithms are particularly useful if the domain of interest does not lie far from the real axis. These methods extend the work of Borwein [1], are easily implemented, and at least in the absence of the precomputation of the Bernoulli numbers, are often superior to commonly used Euler–Maclaurin summation-based methods. We recall that the latter computational methods are implemented in the readily available symbolic computing programs Maple, Mathematica, and Pari [4]. Because of the second complex parameter in the Hurwitz zeta function, special cases of this function lead also to useful results for .j/ .r/ other special functions and numbers, including the polygamma functions , the generalized harmonic numbers Hn , the Euler constant γ , and the Stieltjes constants.
    [Show full text]
  • Expansions of Generalized Euler's Constants Into the Series Of
    Expansions of generalized Euler’s constants into the series of 2 polynomials in π− and into the formal enveloping series with rational coefficients only Iaroslav V. Blagouchine∗ University of Toulon, France. Abstract In this work, two new series expansions for generalized Euler’s constants (Stieltjes constants) γm are 2 obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in π− 2 with rational coefficients and converges slightly better than Euler’s series ∑ n− . The second expan- sion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler’s constants, which is more accurate than several well-known estimations. Finally, in Appendix A, the reader will also find two simple integral definitions for the Stirling numbers of the first kind, as well an upper bound for them. Keywords: Generalized Euler’s constants, Stieltjes constants, Stirling numbers, Factorial coefficients, Series expansion, Divergent series, Semi-convergent series, Formal series, Enveloping series, Asymptotic expansions, Approximations, Bernoulli numbers, Harmonic numbers, Rational coefficients, Inverse pi. I. Introduction and notations I.1. Introduction The ζ-function, which is usually introduced via one of the following series, ∞ 1 > ∑ s , Re s 1 n=1 n arXiv:1501.00740v4 [math.NT] 16 Dec 2016 ζ(s) = (1) ∞ n 1 1 ( 1) − > 1 s ∑ − s , Re s 0, s = 1 1 2 − n=1 n 6 − ∗Corresponding author. Phones: +33–970–46–28–33, +7–953–358–87–23.
    [Show full text]
  • A Theorem for the Closed-Form Evaluation of the First Generalized
    Journal of Number Theory 148 (2015) 537–592 Contents lists available at ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations Iaroslav V. Blagouchine ∗ University of Toulon, France a r t i c l e i n f o a b s t r a c t Article history: Recently, it was conjectured that the first generalized Stieltjes Received 14 January 2014 constant at rational argument may be always expressed Received in revised form 31 July by means of Euler’s constant, the first Stieltjes constant, 2014 the Γ-function at rational argument(s) and some relatively Accepted 19 August 2014 simple, perhaps even elementary, function. This conjecture Available online 7 October 2014 was based on the evaluation of γ (1/2), γ (1/3), γ (2/3), Communicated by David Goss 1 1 1 γ1(1/4), γ1(3/4), γ1(1/6), γ1(5/6), which could be expressed Keywords: in this way. This article completes this previous study and Stieltjes constants provides an elegant theorem which allows to evaluate the Generalized Euler’s constants first generalized Stieltjes constant at any rational argument. Special constants Several related summation formulae involving the first gener- Number theory alized Stieltjes constant and the Digamma function are also Zeta function presented. In passing, an interesting integral representation Gamma function Digamma function for the logarithm of the Γ-function at rational argument is Psi function also obtained. Finally, it is shown that similar theorems may Malmsten be derived for higher Stieltjes constants as well; in particular, Rational arguments for the second Stieltjes constant the theorem is provided in Logarithmic integrals an explicit form.
    [Show full text]
  • An Introduction to the Bernoulli Function
    An Introduction to the Bernoulli Function Peter H. N. Luschny Abstract. We explore a variant of the zeta function interpolating the Bernoulli numbers based on an integral representation suggested by J. Jensen. The Bernoulli function B(s; v) = −s ζ(1 − s; v) can be introduced independently of the zeta function if it is based on a formula first given by Jensen in 1895. We examine the functional equation of B(s; v) and its representation by the Riemann ζ and ξ function, and recast classical results of Hadamard, Worpitzky, and Hasse in terms of B(s; v): The extended Bernoulli function defines the Bernoulli numbers for odd indices basing them on rational numbers studied by Euler in 1735 that underlie the Euler and André numbers. The Euler function is introduced as the difference between values of the Hurwitz-Bernoulli function. The André function and the Seki function are the unsigned versions of the extended Euler resp. Bernoulli function. Notations 3 Index 55 Plots 45 Contents 57 1 – Prologue: extension by interpolation The question André Weil recounts the origin of the gamma function in his historical exposition of number theory [54, p. 275]: “Ever since his early days in Petersburg Euler had been interested in the interpolation of functions and formulas given at first only for integral values of the argument; that is how he had created the theory of the gamma function.” arXiv:2009.06743v2 [math.HO] 29 Sep 2021 The three hundred year success story of Euler’s gamma function shows how fruitful this question is. The usefulness of such an investigation is not limited by the fact that there are infinitely many ways to interpolate a sequence of numbers.
    [Show full text]
  • An Introduction to the Stieltjes Constants
    Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers Volume II(b) Donal F. Connon 18 February 2008 Abstract In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function, including the following: ∞ n 11⎛⎞n kp+1 γ p ()uuk=−⎜⎟(1)log( − + ) ∑∑k pn++11nk==00⎝⎠ x (1)− p+1 γςς()udu=− [(1)pp++ (0,)x (1) (0)] ∫ p p +1 1 ⎛⎞11 2 1⎡ ⎛⎞1⎤ γγ11⎜⎟=−[2 15log 2 − 6 γπγπ log 2] −++−Γ⎢ 4log 2 3log 4log ⎜⎟⎥ ⎝⎠42 2⎣ ⎝⎠4⎦ ∞ ⎡⎤1 1 log log(1/t ) −=dtγγ() u+ 2 γ () u+ γ log u+ log2 u ∫ ⎢⎥u 01 1 ⎣⎦1− tt log(1/ t ) t ∞ n 11⎛⎞n k 221212 ∑∑⎜⎟(−+=−−++ 1)kk log ( 1)γ1 log(2πγ ) πlog (2 π ) nk==00n +12⎝⎠k 242 ∞ n ks∞ −−+1(1)yu n ⎛⎞n xxtuextxt⎛ ⎞ tdusy⎜⎟ s ==2 Φ⎜,,+ 1 ⎟ ∑∑k (ky+−Γ ) (1 t ) ( s ) ∫ ⎡⎤−u (1− t ) (1 − t ) nk==11⎝⎠ 0 11−+()xe t ⎝ ⎠ ⎣⎦ where ()u is the Stieltjes constant and is the Hurwitz-Lerch zeta function. γ p Φ(,,zsu ) Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods. CONTENTS OF VOLUMES I TO VI: Volume/page SECTION: 1. Introduction I/11 2. An integral involving cot x I/17 The Riemann-Lebesgue lemma I/21 3.
    [Show full text]
  • Arxiv:1606.02044V5 [Math.NT] 20 Mar 2018 06.Frayfrhrrfrnet H Aeilpbihdh Published Material Publishe Paper
    Three Notes on Ser’s and Hasse’s Representations for the Zeta-Functions Iaroslav V. Blagouchine1 SeaTech, University of Toulon, France. St. Petersburg State University of Architecture and Civil Engineering, Russia. [email protected], [email protected] Abstract This paper is devoted to Ser’s and Hasse’s series representations for the zeta-functions, as well as to several closely related results. The notes concerning Ser’s and Hasse’s representations are given as theorems, while the related expansions are given either as separate theorems or as formulæ inside the remarks and corollaries. In the first theorem, we show that the famous Hasse’s series for the zeta-function, obtained in 1930 and named after the German mathematician Helmut Hasse, is equivalent to an earlier expression given by a little-known French mathematician Joseph Ser in 1926. In the second theorem, we derive a similar series representation for the zeta-function involving the Cauchy numbers of the second kind (Nørlund numbers). In the third theorem, with the aid of some special polynomials, we generalize the previous results to the Hurwitz zeta-function. In the fourth theorem, we obtain a similar series with Gregory’s coefficients of higher order, which may also be regarded as a functional equation for the zeta-functions. In the fifth theorem, we extend the results of the third theorem to a class of Dirichlet series. As a consequence, we obtain several globally convergent series for the zeta-functions. They are complementary to Hasse’s series, contain the same finite differences and also generalize Ser’s results.
    [Show full text]