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A note on the standard zero-free region for $L$-functions

Sun-Kai Leung

Abstract

In this short note, we establish a standard zero-free region for a general class of $L$-functions for which their logarithms have coefficients with nonnegative real parts, which includes the Rankin--Selberg $L$-functions for unitary cuspidal automorphic representations.

A note on the standard zero-free region for $L$-functions

Abstract

In this short note, we establish a standard zero-free region for a general class of -functions for which their logarithms have coefficients with nonnegative real parts, which includes the Rankin--Selberg -functions for unitary cuspidal automorphic representations.

Paper Structure

This paper contains 3 sections, 1 theorem, 35 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 2.1

Let $L(f,s)$ be an $L$-function in the class above. Suppose $\mathrm{Re}(\Lambda_f(n)) \geqslant 0$ for $n \geqslant 1$ and $L(f,s)$ has at most a simple pole at $s=1.$Note that $L(f,1) \neq 0$ since by assumption $\mathrm{Re} \left(\Lambda_f(n) \right) \geqslant 0$ for $n \geqslant 1.$ Then there e for $s=\sigma+it,$ with the possible exception for a simple real zero $\beta_f<1,$ where $\mathfrak

Theorems & Definitions (7)

  • Remark 2.1
  • Theorem 2.1
  • Example 2.1: Dirichlet $L$-functions
  • Example 2.2: Dedekind zeta functions
  • Example 2.3: Rankin--Selberg $L$-functions
  • Remark 2.2
  • proof : Proof of Theorem \ref{['thm:main']}