Alternating Euler sums at the negative integers

TL;DR

This paper investigates special Dirichlet series related to the Riemann zeta function, extending them to the complex plane and computing their values at negative integers using Bernoulli and Euler numbers.

Contribution

It provides new analytic continuations of three Dirichlet series and explicit formulas for their values at negative integers, connecting them to Bernoulli and Euler numbers.

Findings

Extended the domain of three Dirichlet series to the entire complex plane.

Derived explicit formulas for series values at negative integers.

Connected series values to Bernoulli and Euler numbers.

Abstract

We study three special Dirichlet series, two of them alternating, related to the Riemann zeta function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at poles). These values are given in terms of Bernoulli and Euler numbers.