On Dirichlet Series Involving $\zeta(s)$ and Extensions of the Euler-Mascheroni Constant

TL;DR

This paper introduces a class of Dirichlet series involving the Riemann zeta-function, extends the Euler-Mascheroni constant, and provides explicit evaluations of special values, unifying their analysis within a new framework.

Contribution

It presents a novel class of Dirichlet series, integral representations, and an extended Euler-Mascheroni constant, linking these to special values and number-theoretic constants.

Findings

Explicit integral representations for the series.

Extension of the Euler-Mascheroni constant.

Unified evaluation framework for Dirichlet series.

Abstract

In this paper, we introduce a class of Dirichlet series defined in terms of the Riemann zeta-function, motivated by the study of their special values, and establish integral representations for these series. We also define an extension of the Euler--Mascheroni constant and express certain special values explicitly in terms of the Bendersky constants. These results provide a unified framework for evaluating Dirichlet series involving the Riemann zeta-function at integer arguments, together with the associated number-theoretic constants.