Non-vanishing of Maass form L-functions at the critical point

TL;DR

This paper proves that at least half of the central values of L-functions associated with Maass forms do not vanish as the eigenvalue parameter grows, including results for short intervals, advancing understanding of L-function non-vanishing.

Contribution

It provides the first effective non-vanishing results for a positive proportion of Maass form L-functions at the critical point, including in short intervals.

Findings

At least 50% of L-functions do not vanish at the critical point as eigenvalues grow.

Effective non-vanishing results in short intervals.

Establishment of non-vanishing proportion for Maass form L-functions.

Abstract

In this paper, we consider the family $\{L_j(s)\}_{j=1}^{\infty}$ of $L$-functions associated to an orthonormal basis $\{u_j\}_{j=1}^{\infty}$ of even Hecke-Maass forms for the modular group $SL(2, Z)$ with eigenvalues $\{\lambda_j=\kappa_{j}^{2}+1/4\}_{j=1}^{\infty}$. We prove the following effective non-vanishing result: At least $50 \%$ of the central values $L_j(1/2)$ with $\kappa_j \leq T$ do not vanish as $T\rightarrow \infty$. Furthermore, we establish effective non-vanishing results in short intervals.