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On certain harmonic zeta functions

Mümün Can, Levent Kargın, Mehmet Cenkci, Ayhan Dil

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.

On certain harmonic zeta functions

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.
Paper Structure (9 sections, 21 theorems, 149 equations)

This paper contains 9 sections, 21 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Outline of the study
  3. Proofs of Proposition \ref{['prop-raabe']} and Theorem \ref{['mteo1']}
  4. Proof of Proposition \ref{['prop-raabe']}
  5. Proof of Theorem \ref{['mteo1']}
  6. Laurent coefficients and the values $\zeta_{A\left( k\right) }\left( -2n\right)$
  7. A limit representation for $\gamma_{A\left( k\right) }\left( m\right)$
  8. Two evaluation formulas for $\gamma_{H}\left( n,1/2\right)$
  9. Proof of Theorem \ref{['mteo3']} and its consequences

Key Result

Proposition 1.1

For $k\in\mathbb{N}$, we have

Theorems & Definitions (33)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 3.1
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • ...and 23 more