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

TL;DR

This paper proves that, under the Generalized Riemann Hypothesis, a positive proportion of Dirichlet characters in Galois orbits simultaneously have nonvanishing central L-values when twisted by two fixed characters, extending previous nonvanishing results.

Contribution

It establishes simultaneous nonvanishing of Dirichlet L-functions in Galois orbits under GRH and develops new bounds on mollified moments using Euler product mollifiers and Diophantine approximation techniques.

Findings

Positive proportion of characters with nonvanishing L-values in Galois orbits.

Sharp upper bounds on mollified fourth moments over Galois orbits.

Unconditional second moment calculations for primitive characters in specific 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 $L(1/2,\chi \eta_1) L(1/2,\chi \eta_2) \neq 0$, 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, Mili\'cevi\'c 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.