The Epstein's zeta function is: $Z(Y,s)=\underset{0\neq a\in\mathbb{Z}^{n}}{\sum}(a^{t}Ya)^{-s}$, where Y is a positive symmetric difinite $n \times n$ matrix.
- Why does it converges when $\sigma > \frac{n}{2}$ (where $s=\sigma+it$)?
let $\theta (Y,t)=\underset{0\neq a\in\mathbb{Z}^{n}}{\sum}exp(-\pi (a^{t}Ya)t)$.
how can I proof the following:
$\pi^{-s}\Gamma(s)Z(Y,s)=\frac{1}{2}\int^{\infty}_{0}t^{s-1}[\theta(Y,t)-1]dt$?