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Large deviations for values of $L$-functions attached to cusp forms in the level aspect

Masahiro Mine

Abstract

We study the distribution of values of automorphic $L$-functions in a family of holomorphic cusp forms with prime level. We prove an asymptotic formula for a certain density function closely related to this value-distribution. The formula is applied to estimate large values of $L$-functions.

Large deviations for values of $L$-functions attached to cusp forms in the level aspect

Abstract

We study the distribution of values of automorphic -functions in a family of holomorphic cusp forms with prime level. We prove an asymptotic formula for a certain density function closely related to this value-distribution. The formula is applied to estimate large values of -functions.

Paper Structure

This paper contains 10 sections, 25 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Related results for other zeta and $L$-functions
  4. Preliminary lemmas
  5. Estimates of cumulant-generating functions
  6. Probability density functions
  7. Approximation of the saddle-point
  8. Proof of Theorem \ref{['thm:1.1']}
  9. Corollaries
  10. Comparisons of distribution functions

Key Result

Theorem 1.1

Let $1/2<\sigma \leq1$. For $\tau>0$, we take a real number $\kappa=\kappa(\sigma,\tau)>0$ as the solution of $f'_\sigma(\kappa)=\tau$. Then we obtain uniformly for all $x \in \mathbb{R}$ if $\tau>0$ is large enough, where the implied constant depends only on $\sigma$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • ...and 37 more