Two new proofs of lerch's functional equation

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 00 2 exp[2ninx](n+a)-' n=o are given. The original proof is due to Lerch. If x is real and 0a^ 1, define 00 p(x, a, s) = 2 exp[27Ti'«x](n + a)-s, n=0 where c=Res>l if x is an integer, and cr>0 otherwise. Note that 9?(0, a, s)=£,(s, a), the Hurwitz zeta-function. Furthermore, if a=\, 9?(0, 1, j)=£(s), the Riemann zeta-function. In 1887, Lerch [1] derived the following functional equation for (p(x, a, s).