On some series formed by values of the Riemann Zeta function

TL;DR

This paper generalizes Euler's partial fraction expansion of coth(πz) to new power series involving the Riemann zeta function evaluated at arithmetic sequences and further extends to arbitrary Dirichlet series, introducing novel formulas.

Contribution

It introduces new formulas generalizing classical expansions using the Riemann zeta function and Dirichlet series, not previously documented in existing literature.

Findings

New formulas for series involving the Riemann zeta function

Generalization of Euler's partial fraction expansion

Extension to arbitrary Dirichlet series

Abstract

The partial fraction expansion of coth($\pi$z), due to Euler, is generalized to power series having for coefficients the Riemann zeta function evaluated at certain arithmetic sequences. A further generalization using arbitrary Dirichlet series is also proposed. The resulting formulas are new, as far as we know, since they could not be found in any of the classical or recent handbooks of formulas that were at our disposal.