Zeros of $L$-functions in families near the critical line

TL;DR

This paper develops new methods combining the relative trace formula with analytic techniques to estimate the distribution of zeros of $L$-functions in automorphic families, leading to bounds on their ranks and zero distribution.

Contribution

It introduces a novel approach that merges the relative trace formula with analytic methods to analyze zeros of $L$-functions in automorphic families.

Findings

Established zero density estimates for $L$-functions in various automorphic families.

Derived strong bounds for the average analytic rank at the central point.

Proved average equidistribution of the imaginary parts of zeros.

Abstract

We combine the relative trace formula with analytic methods to obtain zero density estimate for $L$-functions in various families of automorphic representations for $\mathrm{GL}(m)$. Applications include strong bounds for the average analytic rank of these $L$-functions at the central point and average equidistribution results for the imaginary parts of the zeros.