Simultaneous non-vanishing of Dirichlet L-functions

In this paper, we prove the simultaneous non-vanishing of four Dirichlet $L$-functions at any point on the critical line. More precisely, let $\chi_1,\ldots,\chi_4$ be even Dirichlet characters modulo $D_1,\ldots, D_4$ respectively, where the $D_j$ are pairwise co-prime and square-free integers. Under the Generalized Riemann Hypothesis, we prove that $\prod_{j=1}^4 L(1/2+it,\chi \chi_j) \neq 0$ for a positive proportion of Dirichlet characters $\chi \pmod q$, 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 $\chi \pmod q$, though in this case the proportion tends to zero as $q \to \infty$.