On Siegel zeros of Hecke-Landau zeta-functions

Abstract

The primary concern of this paper is to deal with Siegel zeros of Hecke-Landau zeta-functions in an algebraic number field of finite degree over the rationals. As in the rational case with DirichletL-functions, the location of such zeros is closely connected with lower bounds for the corresponding zeta-functions at the points=1. This will be the theme in the first part of the paper. In this second part we first derive a form of the Brun-Titchmarsh theorem in the setting of a number field which is appropriate in our context. Then we turn our attention to the fact that an improvement of the constant in this inequality would lead to the nonexistence of Siegel zeros. The procedure is based on a weighted algebraic form of Selberg's upper bound sieve.