Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Divisor problems for restricted Fourier coefficients of modular forms

Yuk-Kam Lau, Wonwoong Lee

Abstract

Let $d(n)$ be the number of divisors of $n$. We investigate the average value of $d(a_f(p))^r$ for $r$ a positive integer and $a_f(p)$ the $p$-th Fourier coefficient of a cuspidal eigenform $f$ having integral Fourier coefficients, where $p$ is a prime subject to a constraint on the angle associated with the normalized Fourier coefficient.

Divisor problems for restricted Fourier coefficients of modular forms

Abstract

Let be the number of divisors of . We investigate the average value of for a positive integer and the -th Fourier coefficient of a cuspidal eigenform having integral Fourier coefficients, where is a prime subject to a constraint on the angle associated with the normalized Fourier coefficient.

Paper Structure

This paper contains 20 sections, 16 theorems, 160 equations.

Table of Contents

  1. Introduction and results
  2. Introduction
  3. Main result
  4. Auxiliary results
  5. Multivariable Tauberian theorem
  6. Preliminaries for Theorem \ref{['thm_main theorem general']}
  7. Proof of Theorem \ref{['thm_main theorem general']}
  8. Lower bound
  9. Upper bound
  10. Preliminaries on $L$-functions
  11. Automorphic $L$-functions
  12. Artin $L$-functions
  13. $L$-functions of $\mathrm{Sym}^m f$
  14. Rankin-Selberg $L$-functions
  15. $L$-functions of $\mathrm{Sym}^m f \otimes \chi$ and Hypothesis A
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $r \in \mathbb N$ and $f$ be a non-CM newform of weight $k \geq 2$ with integer Fourier coefficients $a_f(n)$, $n\in \mathbb N$. Under Hypothesishypo_nice analytic properties of L-function in § sec_L ftns on sym otimes chi and GRH, as $x \rightarrow \infty$. Moreover, if $I=[0,\pi]$, then the same conclusion holds under GRH only.

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Theorem 1.2
  • Theorem 1.3
  • Remark 3
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 19 more