Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions

被引：29

作者：

Fouvry, Etienne ^{[1
]}

Ganguly, Satadal ^{[2
]}

Kowalski, Emmanuel ^{[3
]}

Michel, Philippe ^{[4
]}

机构：

[1] Univ Paris 11, Math Lab, F-91405 Orsay, France

[2] Indian Stat Inst, Kolkata 700108, India

[3] ETH Zuich D MATH, CH-8092 Zurich, Switzerland

[4] EPFL SB IMB TAN, CH-1015 Lausanne, Switzerland

来源：

COMMENTARII MATHEMATICI HELVETICI | 2014年 / 89卷 / 04期

基金：

欧洲研究理事会;

关键词：

Divisor function; Hecke eigenvalues; Fourier coefficients of modular forms; arithmetic progressions; central limit theorem; Kloosterman sums; monodromy group; Sato-Tate equidistribution;

D O I：

10.4171/CMH/342

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments, and depend crucially on results on the independence of monodromy groups related to products of Kloosterman sums.

引用

页码：979 / 1014

页数：36