Evaluation of Dirichlet series

Abstract

This article addresses the evaluation of Dirichlet series with periodic coefficients. The author establishes necessary and sufficient conditions for expressing a T-periodic Dirichlet series in terms of even and odd sequences, and provides a theorem for computing the values of these series at positive integers, depending on the parity of the coefficient sequence and the exponent. Utilizing the Fourier expansion of Bernoulli polynomials, the article presents explicit formulas and illustrative examples, thereby generalizing classical results related to the Riemann zeta function and Dirichlet L-series