The critical values of generalizations of the Hurwitz zeta function

Abstract

We investigate a few types of generalizations of the Hurwitz zeta function, written $Z(s,a)$ in this abstract, where $s$ is a complex variable and $a$ is a parameter in the domain that depends on the type. In the easiest case we take $a\in \mathbf{R},$ and one of our main results is that $Z(-m,a)$ is a constant times $E_m(a)$ for $0\le m\in \mathbf{Z},$ where $E_m$ is the generalized Euler polynomial of degree $n.$ In another case, $a$ is a positive definite real symmetric matrix of size $n,$ and $Z(-m,a)$ for $0\le m\in \mathbf{Z}$ is a polynomial function of the entries of $a$ of degree $\le mn.$ We will also define $Z$ with a totally real number field as the base field, and will show that $Z(-m,a)\in \mathbf{Q}$ in a typical case.