Laurent expansion of harmonic zeta functions

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Abstract

In this study, we determine certain constants which naturally occur in the Laurent expansion of harmonic zeta functions.

Introduction

Let us consider an analytic function f defined in the half-plane Re(z)>0 and the function ζf defined in a half-plane Re(s)>σ by the Dirichlet series n1f(n)ns whose meromorphic continuation is supposed to have a pole of order m at s=a. The purpose of this study is to examine how the constant Ca involved in the Laurent expansionζf(s)=n=1mbn(sa)n+Ca+O(sa) in a neighborhood of s=a is linked to the sum of the series n1f(n)na in the sense of Ramanujan's summation. To what extent does the knowledge of one enable to determine the other?

In the case where f(n)=1 for all n, ζf is Riemann ζ function. It is well-known that this famous function can be continued as an analytic function in C\{1} and, in a neighborhood of its pole s=1, may be writtenζ(s)=1s1+γ+O(s1), where γ denotes Euler's constantγ=limN{n=1N1nlnN}=0.5772156649 Furthermore, the Ramanujan summation method (cf. [6]) enable to sum the series n11ns for all complex values of s, and the sum of the series at s=1 is nothing else than Euler's constant γ (cf. [6, Eq. (1.24)]). Thus, in this particular simple case, the constant C1 at s=1 and the sum in the sense of Ramanujan of the series at this point are the same.

In the first part of this study, we examine the more difficult case of the analytic function ζH defined for Re(s)>1 byζH(s)=n=1+Hnns, where H=(Hn) is the sequence of harmonic numbers. Apostol-Vu [2] and Matsuoka [12] have shown that this function, called the harmonic zeta function, can be continued as a meromorphic function with a double pole at s=1, and an infinite number of simple poles at s=0 and s=12k for each integer k1. The Ramanujan summation method allows to sum the series n1Hnns for all values of s, which make possible, for each pole s=a, to give an expression of the constant Ca in terms of the sum n1RHnna of the series in the sense of Ramanujan's summation method. Regarding the poles at negative integers, we also find the results previously obtained by Boyadzhiev et al. [4] using a different method. Nevertheless, our method has the advantage of reformulating these results in a more pleasant way, while computing at the same time the value of the constant at s=1 (see formula (6)) that had not been done in [4] nor, as far as we know, in any other article.

In the second part of this paper, we extend our study to a more general class of harmonic zeta functions denoted ζHp for each integer p2, which are defined by the sequence of generalized harmonic numbers Hp=(Hn(p)). This meromorphic function ζHp has simple poles at s=1, and s=mp for m=2,1,0,2,4,6, etc. In the simplest case where p=2, we obtain a complete determination of the constant Ca at the corresponding pole s=a for all values of a (Proposition 3, Proposition 4) as well as a determination of the special values of the function at negative odd integers (Proposition 5). In the general case, we give an expression of the constant C1 at s=1 for all integers p2 (Proposition 6). Finally, we indicate a method to evaluate the constants at the poles s=mp of ζHp based on the one already used in the case p=2.

Section snippets

The harmonic zeta function ζH

If H=(Hn) is the sequence of harmonic numbersHn:=j=1n1j, then, for each integer n1, we recall thatHn=ψ(n+1)+γ, where ψ=Γ/Γ denotes the digamma function (cf. [8, p. 95]).

Definition 1

We call harmonic zeta function and note ζH1 the analytic function defined in the half-plane Re(s)>1 byζH(s)=n=1+Hnns.

Let us remind that the special values of ζH at positive integers are given by Euler's formula (cf. [2], [13]):2ζH(p)=(p+2)ζ(p+1)k=1p2ζ(k+1)ζ(pk)(p2).

The generalized harmonic zeta function ζHp

For each integer p>1, we now consider the sequence Hp=(Hn(p)) of harmonic generalized numbersHn(p):=j=1n1jp. Nota bene. In the remainder of the text, we adopt the lighter notation Hnp in place of Hn(p). To avoid confusions, we will take care to write (Hn)p the p-th power of Hn.

For each integer n1, we recall (cf. [8, p. 95]) thatHnp=ψp(n) where ψp is the analytic function defined in the half-plane Re(x)>1 byψp(x)=ζ(p)+(1)p1(p1)!p1ψ(x+1).

Definition 3

Let p2 be an integer. We call harmonic zeta

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