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A metric approach to zero-free regions for $L$-functions

Nawapan Wattanawanichkul

TL;DR

The paper develops a metric, pretentious framework for zero-free regions of standard and Rankin–Selberg L-functions attached to cuspidal automorphic representations. By replacing the Dirichlet coefficients with $a_{\\pi}(\\mathfrak{n})$ and employing a positive semi-definite family of L-functions in the Lichtman–Pascadi sense, it derives explicit zero-free regions with uniform degree dependence and handles ramified primes via a level-aware distance $\\mathbb{D}^*_{\\sigma}$. The main results give explicit regions in the complex plane where $L(s,\\pi)$ and $L(s,\\(\\pi\\times\\pi'))$ have no zeros (with a small exceptional real zero only in self-dual cases) and establish at-most-one-real-zero statements in self-dual RS settings. The method improves on previous results by reducing degree dependence, avoiding full GRC assumptions, and providing a robust, ramification-tolerant approach that unifies the standard and Rankin–Selberg L-functions in a single analytic framework.

Abstract

For integers $m, m' \ge 1$, let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(m)$ and $\mathrm{GL}(m')$, respectively. We present a new proof of zero-free regions for $L(s, π)$ and for $L(s, π\times π')$ under the assumption that $π, π'$ or $L(s,π\times π')$ is self-dual. Our approach builds on ideas of "pretentious" multiplicative functions due to Granville and Soundararajan (as presented by Koukoulopoulos) and the notion of a positive semi-definite family of automorphic representations due to Lichtman and Pascadi.

A metric approach to zero-free regions for $L$-functions

TL;DR

The paper develops a metric, pretentious framework for zero-free regions of standard and Rankin–Selberg L-functions attached to cuspidal automorphic representations. By replacing the Dirichlet coefficients with and employing a positive semi-definite family of L-functions in the Lichtman–Pascadi sense, it derives explicit zero-free regions with uniform degree dependence and handles ramified primes via a level-aware distance . The main results give explicit regions in the complex plane where and have no zeros (with a small exceptional real zero only in self-dual cases) and establish at-most-one-real-zero statements in self-dual RS settings. The method improves on previous results by reducing degree dependence, avoiding full GRC assumptions, and providing a robust, ramification-tolerant approach that unifies the standard and Rankin–Selberg L-functions in a single analytic framework.

Abstract

For integers , let and be cuspidal automorphic representations of and , respectively. We present a new proof of zero-free regions for and for under the assumption that or is self-dual. Our approach builds on ideas of "pretentious" multiplicative functions due to Granville and Soundararajan (as presented by Koukoulopoulos) and the notion of a positive semi-definite family of automorphic representations due to Lichtman and Pascadi.

Paper Structure

This paper contains 20 sections, 8 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Notation and motivation
  3. Statements of results
  4. Novelty of the present work
  5. Properties of L-functions
  6. Standard L-functions
  7. Rankin--Selberg L-functions
  8. Preliminary computations and useful bounds
  9. Preliminary computations
  10. Useful bounds
  11. Proof of Theorems \ref{['thm1:ZFR-cuspforms']}--\ref{['thm2:ZFR-rankin-selberg']} when N1
  12. Proof of Theorem \ref{['thm1:ZFR-cuspforms']}
  13. Case 1: $\pi$ is not self-dual
  14. Case 2: $\pi$ is self-dual
  15. Proof of Theorem \ref{['thm1:ZFR-rankin-selberg']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $m \ge 1$ be an integer and $\pi \in \mathfrak{F}^*_m$. Let $\sigma,t\in\mathbb{R}$ and $s= \sigma+it$. Then $L(s, \pi)$ has no zero in the region except possibly for one simple real zero, in which case $\pi$ is self-dual. Moreover, when $\pi$ is self-dual, the exponent $m[F:\mathbb{Q}]$ of $3+|t|$ can be reduced to $[F:\mathbb{Q}]$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark
  • Definition 5.1
  • ...and 6 more