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 modulo , a positive proportion of characters modulo in a fixed Galois orbit of primitive characters satisfy the nonvanishing property that , as (with fixed). Previously, only a positive proportion of nonvanishing result was available in Galois orbits (as opposed to simultaneous 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 a Euler product mollifier, and obtaining a lower bound for the mollified second moment, which relies on using results from Diophantine approximation (such as the ‐adic Roth theorem). We also unconditionally compute the second moments for ‐functions associated to primitive Dirichlet characters in full orbits and thinner orbits.