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.

Locations

• Documenta Mathematica - View - PDF