Functional equations for double series of Euler type with coefficients

TL;DR

This paper establishes new functional equations for double Euler-type series with complex coefficients, including a generalization of the Euler double zeta-function and a specific case involving cusp form coefficients, leading to insights on zero divisors.

Contribution

It introduces two novel functional equations for double series of Euler type, extending previous results and applying modular relations for cusp form coefficients.

Findings

Generalized functional equation for Euler double zeta-function

Functional equation for series with cusp form coefficients

Identification of trivial zero divisors

Abstract

We prove two types of functional equations for double series of Euler type with complex coefficients. The first one is a generalization of the functional equation for the Euler double zeta-function, proved in a former work of the second-named author. The second one is more specific, which is proved when the coefficients are Fourier coefficients of cusp forms and the modular relation is essentially used in the course of the proof. As a consequence of functional equation we are able to determine trivial zero divisors.