Multiplicity of nontrivial zeros of primitive L-functions via higher-level correlations

TL;DR

This paper establishes universal bounds on the proportion of nontrivial zeros with specific multiplicities for L-functions linked to automorphic representations, using advanced correlation asymptotics and optimization techniques.

Contribution

It introduces bounds on zero multiplicities of automorphic L-functions by combining higher-level correlation asymptotics with semidefinite programming methods.

Findings

Derived bounds on zero multiplicities for automorphic L-functions.

Applied higher-level correlation asymptotics in a novel way.

Utilized semidefinite programming to optimize bounds.

Abstract

We give universal bounds on the fraction of nontrivial zeros having given multiplicity for L-functions attached to a cuspidal automorphic representation of $\mathrm{GL}_m/\mathbb{Q}$. For this, we apply the higher-level correlation asymptotic of Hejhal and Rudnick & Sarnak in conjunction with semidefinite programming bounds.