# Simultaneous non-vanishing for Dirichlet L-functions

## TL;DR

This paper develops new asymptotic formulas for cubic moments of modular L-functions at the central point, extending previous work on correlations of Hecke eigenvalues and algebraic trace functions, with applications to non-vanishing results of Dirichlet L-functions.

## Contribution

It introduces an asymptotic formula for a generalized cubic moment of modular L-functions, extending prior correlation work and applying it to non-vanishing results for Dirichlet L-functions.

## Key findings

- Established asymptotic formula for cubic moments of modular L-functions.
- Proved positive proportion of characters with non-vanishing L-values.
- Extended correlation techniques to new settings.

## Abstract

We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular L-functions at the central point s = 1/2 and for prime moduli q. As an application, we exploit our recent result on the mollification of the fourth moment of Dirichlet L-functions to derive that for any pair $(\omega_1,\omega_2)$ of multiplicative characters modulo q, there is a positive proportion of $\chi$ (mod q) such that $L(\chi, 1/2 ), L(\chi\omega_1, 1/2 )$ and $L(\chi\omega_2, 1/2)$ are simultaneously not too small.