On sums involving powers of harmonic numbers

TL;DR

This paper investigates Dirichlet series generated by powers of harmonic numbers, exploring their analytic properties, asymptotics, and potential implications for understanding the irrationality of the Euler-Mascheroni constant.

Contribution

It introduces new Dirichlet series related to harmonic numbers and analyzes their properties, offering insights into their behavior and connections to fundamental constants.

Findings

Derived series involving harmonic numbers and their properties

Analyzed the behavior of Dirichlet series at negative integers and poles

Proposed links between analytic properties and the irrationality of the Euler-Mascheroni constant

Abstract

In this paper, we study a Dirichlet series generated by powers of harmonic numbers. As an application of these functions, we derive certain series involving harmonic numbers. We also study the analytic properties of these Dirichlet series such as values negative integers and behavior at poles. In particular, objects similar to the Stieltjes constants are discussed. Asymptotics of the sums involving harmonic numbers are also studied. From these results I showed a connection between its analytic properties and a possible route to showing the irrationality of the Euler-Mascheroni constant.