View source for Hurwitz zeta function


''generalised zeta function''

An [[Dirichlet series]] related to the [[Riemann zeta function]] which may be used to exhibit properties of various [[Dirichlet L-function]]s.

The Hurwitz zeta function $\zeta(\alpha,s)$ is defined for real $\alpha$, $0 < \alpha \le 1$ as
$$
\zeta(\alpha,s) = \sum_{n=0}^\infty (n+\alpha)^{-s} \ .
$$
The series is convergent, and defines an [[analytic function]], for $\Re s > 1$.  The function possesses an [[analytic continuation]] to the whole $s$-plane except for a simple pole of residue 1 at $s=1$.

==References==
* Tom M. Apostol, "Introduction to Analytic Number Theory", Undergraduate Texts in Mathematics, Springer (1976) {{ISBN|0-387-90163-9}} {{ZBL|0335.10001}}

{{TEX|done}}