BILATERAL ZETA FUNCTIONS AND THEIR APPLICATIONS

Abstract

We introduce a new type of multiple zeta function, which we call a bilateral zeta function. 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 for the Dedekind η-function, Ramanujan's formula, Fourier expansion of the Barnes zeta function and multiple Iseki's formula.