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

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.