Simultaneous non-vanishing of Dirichlet L-functions

Abstract

In this paper, we prove the simultaneous non-vanishing of four Dirichlet $L$-functions at any point on the critical line. More precisely, let $\ddot{I}{‡}_1,\dots ,\ddot{I}{‡}_4$ be even Dirichlet characters modulo $D_1,\dots ,D_4$ respectively, where the $D_j$ are pairwise co-prime and square-free integers. Under the Generalized Riemann Hypothesis, we prove that ${\prod }_{j=1}^{4}L(1/2+it,\ddot{I}‡\ddot{I}{‡}_j)\ne 0$ for a positive proportion of Dirichlet characters $\ddot{I}‡(\mathrm{mod}q)$, with $q$ prime and sufficiently large in terms of the $D_j$ and $t$ (and with an explicit relationship between $D_j,t$ and $q$). Unconditionally, we also prove a simultaneous non-vanishing result for four Dirichlet $L$-functions for infinitely many characters $\ddot{I}‡(\mathrm{mod}q)$, though in this case the proportion tends to zero as $q\to \infty$.

63 pages