A Note on Some Constants Related to the Zeta-Function and Their Relationship with the Gregory Coefficients
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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. -
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 ). -
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]. -
Arxiv:1408.3902V9 [Math.NT] 30 May 2016 Insadfrtecuh Ubr Ftescn Kind
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. Abstract In this paper, two new series for the logarithm of the Γ-function are presented and studied. Their polygamma analogs are also obtained and discussed. These series involve the Stirling numbers of the first kind and have the property to contain only rational coefficients for certain arguments related to 1 Γ 1 1 Ψ 1 1 Ψ π− . In particular, for any value of the form ln ( 2 n απ− ) and k( 2 n απ− ), where k stands for ± 1 ± the kth polygamma function, α is positive rational greater than 6 π, n is integer and k is non-negative integer, these series have rational terms only. In the specified zones of convergence, derived series con- m 2 verge uniformly at the same rate as ∑(n ln n)− , where m = 1,2,3,..., depending on the order of the polygamma function. Explicit expansions into the series with rational coefficients are given for the Γ 1 Γ 1 Γ 1 1 Ψ 1 Ψ 1 1 Ψ 1 most attracting values, such as ln (π− ), ln (2π− ), ln ( 2 + π− ), (π− ), ( 2 + π− ) and k(π− ). Besides, in this article, the reader will also find a number of other series involving Stirling numbers, Gre- gory’s coefficients (logarithmic numbers, also known as Bernoulli numbers of the second kind), Cauchy numbers and generalized Bernoulli numbers. Finally, several estimations and full asymptotics for Gre- gory’s coefficients, for Cauchy numbers, for certain generalized Bernoulli numbers and for certain sums with the Stirling numbers are obtained. -
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. -
A3 Integers 18A (2018) Three Notes on Ser's and Hasse's
#A3 INTEGERS 18A (2018) THREE NOTES ON SER’S AND HASSE’S REPRESENTATIONS FOR THE ZETA-FUNCTIONS Iaroslav V. Blagouchine1 SeaTech, University of Toulon, France and St. Petersburg State University of Architecture and Civil Engineering, Russia [email protected]; [email protected] Received: 5/9/17, Accepted: 3/2/18, Published: 3/16/18 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 gen- eralize 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 conse- quence, we obtain several globally convergent series for the zeta-functions. -
The Sum of the Approximation Errors of Harmonic Numbers
Appendix A The Sum of the Approximation Errors of Harmonic Numbers A.1 Introduction In this appendix, we compile a list of identities that involve the sum of approxima- tion errors of harmonic numbers by the natural logarithmic function. Specifically, a famous result, due to Euler, is that the limit: ˚ Xn 1 « lim log n n!1 k kD1 exists and is given by a constant D 0:55721. This was proved in Chap. 2. Consequently, we know that the following alternating series: X1 k .1/ Hk log k (A.1.1) kD1 converges to some finite value. Our first goal is to derive an exact expression to this value. Using the Euler-Maclaurin summation formula, it can be shown that the approx- imation error Hn log n has the asymptotic expansion: 1 1 X B H log n k n 2n knk kD2 P 1 Œ This shows that the sum of approximation errors kD1 Hk log k diverges. Adding an additional term from the asymptotic expansion above gives us: 1 H log n O.1=n2/ n 2n © Springer International Publishing AG 2018 151 I. M. Alabdulmohsin, Summability Calculus, https://doi.org/10.1007/978-3-319-74648-7 152 A The Sum of the Approximation Errors of Harmonic Numbers Hence, the following series: X1 1 H log k (A.1.2) k 2k kD1 converges to some finite value. Our second goal is to derive a closed-form expression for this series. Finally, using the Euler-Maclaurin summation formula, Cesaro and Ramanujan, among others, deduced the following asymptotic expansion [Seb02]: p 1 H log n.n C 1/ n 6n.n C 1/ As a result, the natural logarithms of the geometric means offer a more accurate approximation to the harmonic numbers. -
New Series Identities with Cauchy, Stirling, and Harmonic Numbers, and Laguerre Polynomials
July 16, 2020 New series identities with Cauchy, Stirling, and harmonic numbers, and Laguerre polynomials Khristo N. Boyadzhiev Ohio Northern University Department of Mathematics and Statistics Ada, Ohio 45810, USA e-mail: [email protected] Abstract In this article we use an interplay between Newton series and binomial formulas in order to generate a number of series identities involving Cauchy numbers (also known as Gregory coefficients or Bernoulli numbers of the second kind), harmonic numbers, Laguerre polynomials, and Stirling numbers of the first kind. 2010 Mathematics Subject Classification: Primary 11B65; Secondary, 05A19, 33B99, 30B10. Keywords: Cauchy number, harmonic number, Stirling number, Newton series, binomial identity, Riemann’s zeta function, Laguerre polynomial, series with special numbers. 1. Introduction The Newton interpolation series is a classic tool in analysis with important applications ([12], [17], [18], [19]). In this paper, we use the Newton series in order to construct two summation formulas; the first one involving Cauchy numbers and the other one - Stirling numbers of the first kind. We obtain a number of series identities - closed form evaluations of several series involving Cauchy numbers together with harmonic numbers. We also present similar series with harmonic numbers and Stirling numbers of the first kind. The results are close to some classical results about such series. The Cauchy numbers cn (see OEIS A006232 and A006233) are defined by the generating function x ∞ c = ∑ n xxn (| |< 1) ln(xn+ 1)n=0 ! 1 and they can also be defines as an integral of the falling factorial (see equation (2) below). These numbers are called Cauchy numbers of the first kind by Comtet [11]. -
GLOBAL SERIES for ZETA FUNCTIONS 1. Introduction The
GLOBAL SERIES FOR ZETA FUNCTIONS PAUL THOMAS YOUNG Abstract. We provide two general families of everywhere-convergent series expansions for Barnes multiple zeta functions involving Bernoulli polynomials of the second kind and weigh- ted Stirling numbers. These contain the classical results of Ser and Hasse and several recent generalizations as special cases. We also show how these series have good p-adic analogues. 1. Introduction The values of the Riemann zeta function ζ(s) at positive integers k may be interpreted as the (reciprocals of) probabilities that a set of k \randomly chosen" positive integers is relatively prime; this is a consequence of the celebrated Euler product 1 X Y ζ(s) := n−s = (1 − p−s)−1 (<(s) > 1) (1.1) n=1 primes p observed for real values of s by Euler as an expression of the Fundamental Theorem of Arith- metic. Upon analytic continuation, the values of ζ(s) at the negative integers are rational numbers whose numerators are related to the class numbers of cyclotomic fields. And of course the behavior of ζ(s) in the critical strip 0 < <(s) < 1 is intimately connected to questions concerning the distribution of prime numbers. Since zeta functions reveal different arithmetic information depending on where they are evaluated, there is some philosophical interest in expressions for them which are valid on the entire complex plane. Series expansions of this type were given by Ser [17] and by Hasse [9] for the Hurwitz zeta function, and several generalizations of these results are also known. In this article we give two very general families (Theorems 3.1 and 3.2) of series expansions for multiple zeta functions which unify many of the known series, such as those in [3, 17, 9].