Bilateral zeta functions and their applications

TL;DR

This paper introduces bilateral zeta functions, a new class of multiple zeta functions, and demonstrates their usefulness in deriving classical formulas through their Fourier series expansion and relation to Barnes zeta functions.

Contribution

The paper defines bilateral zeta functions, explores their properties, and shows how they simplify proofs of important formulas in special functions and number theory.

Findings

Bilateral zeta functions are periodic and have a Fourier series expansion.

Barnes zeta functions can be expressed as finite sums of bilateral zeta functions.

New proofs of reflection formulas, inversion formulas, and Fourier expansions are obtained.

Abstract

We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta function. Especially, we prove that the bilateral zeta function has a nice Fourier series expansion and the Barnes zeta function can be expressed as a finite sum of bilateral zeta functions. By these properties of the bilateral zeta functions, We obtain simple proofs of some formulas, for example the reflection formula for the multiple gamma function, the inversion formula of the Dedekind eta function, Ramanujan's formula, Fourier expansion of the Barnes zeta function and multiple Iseki's formula.