Non-vanishing of Dirichlet L-functions at the central point

Abstract

This paper deals with the matter of the non-vanishing of Dirichlet L-functions at the central point for all primitive characters χ. More precisely, S. Chowla conjectured that , but this remains still unproved. We first give an efficient algorithm to compute the order n χ of zero of L(s,χ) at . This enables us to efficiently compute n χ for L-functions with very large conductor near 1016. Then, we prove that for all real characters χ of modulus less than 10 10. Finally we give some estimates for n χ and the lowest zero of L(s,χ) on the critical line in terms of the conductor q. © 2008 Springer-Verlag Berlin Heidelberg.