On certain harmonic zeta functions

TL;DR

This paper investigates harmonic zeta functions, deriving their values at specific points, Laurent expansions, and closed-form expressions, enhancing understanding of their properties and relationships with other special functions.

Contribution

It introduces new results on harmonic zeta functions, including explicit values at negative even integers, Laurent expansions, and closed-form evaluations at positive odd integers.

Findings

Values at negative even integers are determined.

Laurent expansions at poles are described.

Values at positive odd integers are expressed in closed form.

Abstract

This study deals with certain harmonic zeta functions, one of them occurs in the study of the multiplication property of the harmonic Hurwitz zeta function. The values at the negative even integers are found and Laurent expansions at poles are described. Closed-form expressions are derived for the Stieltjes constants that occur in Laurent expansions in a neighborhood of s=1. Moreover, as a bonus, it is obtained that the values at the positive odd integers of three harmonic zeta functions can be expressed in closed-form evaluations in terms of zeta values and log-sine integrals.