On an error term for the first moment of twisted $L$-functions

TL;DR

This paper improves the error term in the first moment of twisted L-functions involving a Maass cusp form and primitive Dirichlet characters, leading to results on non-vanishing of these L-functions at the critical line.

Contribution

It provides a sharper error estimate for the first moment of twisted L-functions and demonstrates the existence of non-vanishing L-values for large prime moduli.

Findings

Enhanced error bounds for the first moment of twisted L-functions.

Proof of non-vanishing of L-functions at the critical line for large primes.

Explicit growth condition on prime q for non-vanishing results.

Abstract

Let $f$ be a Hecke-Maass cusp form for the full modular group and let $\chi$ be a primitive Dirichlet character modulo a prime $q$. Let $s_0=\sigma_0+it_0$ with $\frac{1}{2}\leq\sigma_0<1$. We improve the error term for the first moment of $L(s_0,f\otimes\chi)\overline{L(s_0,\chi)}$ over the family of even primitive Dirichlet characters. As an application, we show that for any $t\in\mathbb{R}$, there exists a primitive Dirichlet character $\chi$ modulo $q$ for which $L(1/2+it,f\otimes\chi)L(1/2+it,\chi)\neq0$ if the prime $q$ satisfies $q\gg (1+|t|)^{\frac{543}{25}+\varepsilon}$.