An explicit expression of the Lerch zeta function on maximal domains of holomorphy

TL;DR

This paper derives explicit formulas for the Lerch zeta function's analytic continuation across maximal domains and relates its functional equation to Apostol's, advancing understanding of its complex properties.

Contribution

Provides explicit expressions for the Lerch zeta function's continuation and links its functional equation to Apostol's, enhancing theoretical understanding.

Findings

Explicit formulas for maximal domain holomorphy

Extended formulas for special values at non-positive integers

Connection between Lerch's and Apostol's functional equations

Abstract

We give two results on the Lerch zeta function $\Phi(z,\,s,\,w)$. The first is to give an explicit expression providing both the analytic continuation of $\Phi$ in $n$-variables $(n \in \{1,\,2,\,3\})$ to maximal domains of holomorphy in $\mathbb{C}^n$ with computable evaluation and an extended formula for the special values of $\Phi$ at non-positive integers in the variable $s$. The second is to show that Lerch's functional equation is essentially the same as Apostol's functional equation using the first result.