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

Felipe Gonçalves, David de Laat, Nando Leijenhorst

TL;DR

This work bounds the fraction of nontrivial zeros of $L(s,\pi)$ with prescribed multiplicity for irreducible cuspidal automorphic representations $\pi$ of $\mathrm{GL}_m/\mathbb{Q}$ by exploiting the $n$-level correlations of zeros via Hejhal and Rudnick–Sarnak, together with semidefinite programming. The authors derive an optimization framework where the asymptotic ratio $\liminf_{T\to\infty} \frac{Z_n(T)}{N(T)}$ is bounded below by $1 - \frac{c_{n,m}}{(n-1)!}$, with $c_{n,m}$ obtained from a kernel- and reproducing-kernel-based SDP. They provide an analytic solution for $n=2$ yielding an explicit bound, and develop polynomial- and shift-based parametrizations to compute bounds for larger $n$ (notably $(n,m)=(3,1)$, $(3,2)$, and $(4,1)$). The results deliver new universal lower bounds on the multiplicity distribution of zeros, demonstrating a scalable framework that blends harmonic analysis, representation theory, and convex optimization to study zero statistics of automorphic $L$-functions.

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.

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

TL;DR

This work bounds the fraction of nontrivial zeros of with prescribed multiplicity for irreducible cuspidal automorphic representations of by exploiting the -level correlations of zeros via Hejhal and Rudnick–Sarnak, together with semidefinite programming. The authors derive an optimization framework where the asymptotic ratio is bounded below by , with obtained from a kernel- and reproducing-kernel-based SDP. They provide an analytic solution for yielding an explicit bound, and develop polynomial- and shift-based parametrizations to compute bounds for larger (notably , , and ). The results deliver new universal lower bounds on the multiplicity distribution of zeros, demonstrating a scalable framework that blends harmonic analysis, representation theory, and convex optimization to study zero statistics of automorphic -functions.

Abstract

We give universal bounds on the fraction of nontrivial zeros having given multiplicity for L-functions attached to a cuspidal automorphic representation of . For this, we apply the higher-level correlation asymptotic of Hejhal and Rudnick & Sarnak in conjunction with semidefinite programming bounds.
Paper Structure (7 sections, 4 theorems, 94 equations, 2 figures, 2 tables)

This paper contains 7 sections, 4 theorems, 94 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Derivation of the optimization problem
  3. Analytic solution for n=2
  4. Symmetry and rank-1 structure
  5. Parametrization using polynomials
  6. Parametrization using shifts
  7. The Fourier transform of Hn-1

Key Result

Theorem 1

Let $L(s,\pi)$ be the $L$-function attached to an irreducible cuspidal automorphic representation $\pi$ of $\mathrm{GL}_m/\mathbb{Q}$. Assuming GRH for $L(s,\pi)$ we have where the results for $(n,m)=(3,2), (4,1)$ hold under the additional assumption that certain series of rational functions are summed correctly using Maple; see Section sec:setup.

Figures (2)

  • Figure 1: The hexagon $H_2$ and the square $C$ (shaded).
  • Figure 2: The possible shapes of the intersection of the supports of $g_i(u_1, u_2)$ and $g_{i'}(u_1+x, u_2-x)$ for $0 \leq x \leq 1/2$ (left) and $1/2 \leq x \leq 1$ (right).

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof