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 ∈ R, and one of our main results is that Z(−m, a) is a constant times Em(a) for 0 ≤ m ∈ Z, where Em 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 ≤ m ∈ Z is a polynomial function of the entries of a of degree ≤ mn. We will also define Z with a totally real number field as the base field, and will show that Z(−m, a) ∈ Q in a typical case. 2010 Mathematics Subject Classification: 11B68, 11M06, 30B50, 33E05. Introduction This paper is divided into four parts. In the first part we consider a generalization of Hurwitz zeta function given by (0.1) ζ(s; a, γ) = ∞