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

Valentin Blomer, Jesse Thorner

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.

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

Abstract

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

Paper Structure

This paper contains 14 sections, 19 theorems, 119 equations.

Table of Contents

  1. Introduction and statement of results
  2. Zero density estimates
  3. Applications
  4. Order of vanishing at $s=1/2$
  5. Equidistribution of fractional imaginary parts of nontrivial zeros
  6. Properties of L-functions
  7. Large sieve inequalities
  8. Proof of Theorems \ref{['thm:ZDE']} and \ref{['thm:ZDE2']}
  9. Proof of \ref{['thm:ZDE']}
  10. Proof of \ref{['thm:ZDE2']}
  11. Proof of \ref{['thm:order_of_vanishing']}
  12. Proofs of \ref{['thm:zeros_equidistr', 'thm:zeros_equidistr2']}
  13. Proof of \ref{['thm:zeros_equidistr']}
  14. Proof of \ref{['thm:zeros_equidistr2']}

Key Result

Theorem 1.1

Let $q\geq 2$ be an integer, $T\geq 2$, and $\sigma\geq 1/2$. If $0<B<1$, then The function $F(q)$ is defined in eqn:fq below and satisfies $F(q) \ll 2^{\omega(q)}$, where $\omega(q)$ is the number of distinct prime factors of $q$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 3.1
  • ...and 24 more