#### DOCUMENTA MATHEMATICA, Vol. 15 (2010), 489-506

Goro Shimura

The Critical Values of Generalizations of the Hurwitz Zeta Function

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 $a\in\R,$ and one of our main results is that $Z(-m,a)$ is a constant times $E_m(a)$ for $0le m\in\Z,$ 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 $0le m\in\Z$ 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 $Z(-m,a)\in\Q$ in a typical case.

2010 Mathematics Subject Classification: 11B68, 11M06, 30B50, 33E05.

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