Simultaneous nonvanishing of Dirichlet $L$-functions in Galois orbits

Abstract

Under the Generalized Riemann Hypothesis, we prove that given any two distinct imprimitive Dirichlet characters ${\eta }_1,{\eta }_2$ modulo $q=p^k$, a positive proportion of characters $\chi$ modulo $q$ in a fixed Galois orbit of primitive characters satisfies the nonvanishing property that $L(1/2,\chi {\eta }_1)L(1/2,\chi {\eta }_2)\ne 0$, as $k\to \infty$ (with $p$ fixed). Previously, only a positive proportion of nonvanishing result was available in Galois orbits (as opposed to simultaneously nonvanishing), due to work of Khan, Milićević and Ngo. The main ingredients are obtaining a sharp upper bound on the mollified fourth moment over the Galois orbit using an Euler product mollifier, and obtaining a lower bound for the mollified second moment, which relies on using results from Diophantine approximation (such as the $p$-adic Roth theorem). We also unconditionally compute the second moments for $L$--functions associated to primitive Dirichlet characters in full orbits and thinner orbits.