Two new proofs of Lerch’s functional equation

Abstract

One bright Sunday morning I went to church, And there I met a man named Lerch. We both did sing in jubilation, For he did show me a new equation. Two simple derivations of the functional equation of \[ ∑ n = 0 ∞ exp ⁡ [ 2 π i n x ] ( n + a ) − s \sum \limits _{n = 0}^\infty {\exp [2\pi inx]{{(n + a)}^{ - s}}} \] are given. The original proof is due to Lerch.