The critical values of generalizations of the Hurwitz zeta function

  • Goro Shimura

    Department of Mathematics Princeton University Princeton NJ 08544-1000 USA
The critical values of generalizations of the Hurwitz zeta function cover
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Abstract

We investigate a few types of generalizations of the Hurwitz zeta function, written Z(s,a)Z(s,a) in this abstract, where ss is a complex variable and aa is a parameter in the domain that depends on the type. In the easiest case we take aR,a\in\R, and one of our main results is that Z(m,a)Z(-m,a) is a constant times Em(a)E_m(a) for 0lemZ,0le m\in\Z, where EmE_m is the generalized Euler polynomial of degree n.n. In another case, aa is a positive definite real symmetric matrix of size n,n, and Z(m,a)Z(-m,a) for 0lemZ0le m\in\Z is a polynomial function of the entries of aa of degree lemn.le mn. We will also define ZZ with a totally real number field as the base field, and will show that Z(m,a)\QZ(-m,a)\in\Q in a typical case.

Cite this article

Goro Shimura, The critical values of generalizations of the Hurwitz zeta function. Doc. Math. 15 (2010), pp. 489–506

DOI 10.4171/DM/303