A signed analog of Euler's reduction formula for the double zeta function

TL;DR

This paper introduces an elementary proof for a signed analog of Euler's reduction formula for the double zeta function, extending Euler's classical results to include sign variations, with implications for evaluating these functions.

Contribution

It provides a new elementary proof of a signed reduction formula for the double zeta function, generalizing Euler's original evaluation to include sign variations.

Findings

Elementary proof of the signed reduction formula

Extension of Euler's evaluation to signed double zeta functions

Connection to previous contour integration proofs

Abstract

The double zeta function is a function of two arguments defined by a double Dirichlet series, and was first studied by Euler in response to a letter from Goldbach in 1742. By calculating many examples, Euler inferred a closed form evaluation of the double zeta function in terms of values of the Riemann zeta function, in the case when the two arguments are positive integers with opposite parity. Here, we consider a signed analog of Euler's evaluation: namely a reduction formula for the signed double zeta function that reduces to Euler's evaluation when the signs are specialized to 1. This formula was first stated in a 1997 paper by Borwein, Bradley and Broadhurst and was subsequently proved by Flajolet and Salvy using contour integration. The purpose here is to give an elementary proof based on a partial fraction identity.