HurwitzZetaFunktion

The Hurwitz zeta function is a function of two complex variables, usually denoted by $\zeta(s, q)$. It is a generalization of the Riemann zeta function. The function is defined by the series $\zeta(s, q) = \sum_{n=0}^{\infty} \frac{1}{(n+q)^s}$ for complex numbers $s$ and $q$, where the real part of $s$ is greater than 1, and $q$ is not a non-positive integer. The series converges uniformly in any compact region within its domain of definition.

The Hurwitz zeta function has an analytic continuation to the entire complex $s$-plane, except for a simple pole at $s=1$. The residue at this pole is 1. When $q=1$, the Hurwitz zeta function reduces to the Riemann zeta function: $\zeta(s, 1) = \zeta(s)$.

The Hurwitz zeta function appears in various areas of mathematics and physics, including number theory, analytic number theory, and quantum field theory. It is related to other special functions, such as the polylogarithm and the Lerch transcendent. Integral representations and functional equations exist for the Hurwitz zeta function, which are useful for its study and computation. For example, a common integral representation is $\zeta(s, q) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{x^{s-1} e^{-qx}}{1-e^{-x}} dx$ for $\operatorname{Re}(s) > 1$ and $\operatorname{Re}(q) > 0$.