On the Non-Trivial Zeros off the Critical Line for L-functions from the Extended Selberg Class

Abstract

The principal problem considered in this paper can be formulated as follows: given an L-function satisfying the Riemann Hypothesis or at least a non-trivial density estimate, is it true that it has an Euler product expansion? A positive answer would mean that arithmetic is necessary for proving the Riemann Hypothesis or a non-trivial density estimate, respectively. The paper contains a solution for degree one L-functions from the extended Selberg class. The main tools in the proof are: explicit description of the structure of the extended Selberg class in degree one due to A. Perelli and the first named author (see [7]) and a “hybrid” type joint universality theorem for Dirichlet L-functions. The latter result seems to be of an independent interest, embodying in one statement a special case of the classical Kronecker-Weyl theorem on diophantine approximations and Voronin’s joint universality theorem for Dirichlet L-functions.