On Dirichlet Series Involving $\zeta (s)$ 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.