The critical values of generalizations of the Hurwitz zeta function (Q612850)

Summary: 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 \(a\in\mathbb R,\) and one of our main results is that \(Z(-m,a)\) is a constant times \(E_m(a)\) for \(0\leq m\in\mathbb Z,\) 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 \(0\leq m\in\mathbb Z\) is a polynomial function of the entries of \(a\) of degree \(\leq mn.\) We also define \(Z\) with a totally real number field as the base field, and show that \(Z(-m,a)\in\mathbb Q\) in a typical case.