Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a level analog of Selberg's result on $S(t)$

Qingfeng Sun, Hui Wang, Yanxue Yu

TL;DR

This work establishes an unconditional level-aspect analogue of Selberg's results for S(t) in the GL2 setting of holomorphic weight-2 cusp forms at prime level q. It develops a precise truncated Dirichlet-series approximation for S(t,f) and leverages a weighted zero-density estimate to handle the family of L-functions, enabling exact moment asymptotics. The authors show that the even moments of S(t,f) scale like (log log q)^{n/2} with explicit constants and that odd moments vanish, leading to a weighted central limit theorem. Consequently, the distribution of S(t,f) normalized by √log log q converges to a Gaussian with mean 0 and variance 1/(2π^2) under the harmonic weight measure, revealing a level-aspect probabilistic behavior analogous to Selberg's classical results.

Abstract

Let $S(t,f)=π^{-1}\arg L(1/2+it, f)$, where $f$ is a holomorphic Hecke cusp form of weight $2$ and prime level $q$. In this paper, we establish an unconditional asymptotic formula for the moments of $S(t,f)$, providing a level aspect analogue of Selberg's classical work on $S(t)$. As a consequence, we derive a weighted central limit theorem for the distribution of $S(t,f)$ normalized by $\sqrt{\log\log q}$. To this end, we develop a precise approximation for $S(t,f)$ via a truncated Dirichlet series and employ a weighted zero-density estimate for the corresponding family of $L$-functions.

On a level analog of Selberg's result on $S(t)$

TL;DR

This work establishes an unconditional level-aspect analogue of Selberg's results for S(t) in the GL2 setting of holomorphic weight-2 cusp forms at prime level q. It develops a precise truncated Dirichlet-series approximation for S(t,f) and leverages a weighted zero-density estimate to handle the family of L-functions, enabling exact moment asymptotics. The authors show that the even moments of S(t,f) scale like (log log q)^{n/2} with explicit constants and that odd moments vanish, leading to a weighted central limit theorem. Consequently, the distribution of S(t,f) normalized by √log log q converges to a Gaussian with mean 0 and variance 1/(2π^2) under the harmonic weight measure, revealing a level-aspect probabilistic behavior analogous to Selberg's classical results.

Abstract

Let , where is a holomorphic Hecke cusp form of weight and prime level . In this paper, we establish an unconditional asymptotic formula for the moments of , providing a level aspect analogue of Selberg's classical work on . As a consequence, we derive a weighted central limit theorem for the distribution of normalized by . To this end, we develop a precise approximation for via a truncated Dirichlet series and employ a weighted zero-density estimate for the corresponding family of -functions.
Paper Structure (4 sections, 15 theorems, 117 equations)

This paper contains 4 sections, 15 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An approximation of $S(t,f)$
  4. Proof of Theorems \ref{['main-theorem']} and \ref{['main-corollary']}

Key Result

Theorem 1.1

Let $t>0$ and $n\in \mathbb{N}$ be given. For sufficiently large prime number $q$, we have where the harmonic weight $\omega_f$ is defined by the harmonic weight below and $C_n$ is defined by

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • ...and 15 more