Laurent expansion of harmonic zeta functions
Introduction
Let us consider an analytic function f defined in the half-plane and the function defined in a half-plane by the Dirichlet series whose meromorphic continuation is supposed to have a pole of order m at . The purpose of this study is to examine how the constant involved in the Laurent expansion in a neighborhood of is linked to the sum of the series in the sense of Ramanujan's summation. To what extent does the knowledge of one enable to determine the other?
In the case where for all n, is Riemann ζ function. It is well-known that this famous function can be continued as an analytic function in and, in a neighborhood of its pole , may be written where γ denotes Euler's constant Furthermore, the Ramanujan summation method (cf. [6]) enable to sum the series for all complex values of s, and the sum of the series at is nothing else than Euler's constant γ (cf. [6, Eq. (1.24)]). Thus, in this particular simple case, the constant at 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 defined for by where 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 , and an infinite number of simple poles at and for each integer . The Ramanujan summation method allows to sum the series for all values of s, which make possible, for each pole , to give an expression of the constant in terms of the sum 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 (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 for each integer , which are defined by the sequence of generalized harmonic numbers . This meromorphic function has simple poles at , and for , etc. In the simplest case where , we obtain a complete determination of the constant at the corresponding pole 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 at for all integers (Proposition 6). Finally, we indicate a method to evaluate the constants at the poles of based on the one already used in the case .
Section snippets
The harmonic zeta function
If is the sequence of harmonic numbers then, for each integer , we recall that where denotes the digamma function (cf. [8, p. 95]).
Definition 1 We call harmonic zeta function and note 1 the analytic function defined in the half-plane by
The generalized harmonic zeta function
For each integer , we now consider the sequence of harmonic generalized numbers Nota bene. In the remainder of the text, we adopt the lighter notation in place of . To avoid confusions, we will take care to write the p-th power of .
For each integer , we recall (cf. [8, p. 95]) that where is the analytic function defined in the half-plane by Definition 3 Let be an integer. We call harmonic zeta
References (14)
- et al.
Dirichlet series related to the Riemann zeta function
J. Number Theory
(1984) A note on some alternating series involving zeta and multiple zeta values
J. Math. Anal. Appl.
(2019)A formula of S. Ramanujan
J. Number Theory
(1987)- et al.
Ramanujan's master theorem
Ramanujan J.
(2012) A special constant and series with zeta values and harmonic numbers
Gaz. Mat., Ser. A
(2018)- et al.
The values of an Euler sum at the negative integers and a relation to a certain convolution of Bernoulli numbers
Bull. Korean Math. Soc.
(2008) - et al.
The power series coefficients of functions defined by Dirichlet series
Ill. J. Math.
(1961)