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Commentarii Mathematici Helvetici


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Volume 89, Issue 4, 2014, pp. 979–1014
DOI: 10.4171/CMH/342

Published online: 2014-11-25

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

Étienne Fouvry[1], Satadal Ganguly[2], Emmanuel Kowalski[3] and Philippe Michel[4]

(1) Université Paris Sud, Orsay, France
(2) Indian Statistical Institute, Kolkata, India
(3) ETH Zürich, Switzerland
(4) Ecole Polytechnique Fédérale de Lausanne, Switzerland

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.

Keywords: Divisor function, Hecke eigenvalues, Fourier coefficients of modular forms, arithmetic progressions, central limit theorem, Kloosterman sums, monodromy group, Sato–Tate equidistribution

Fouvry Étienne, Ganguly Satadal, Kowalski Emmanuel, Michel Philippe: Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions. Comment. Math. Helv. 89 (2014), 979-1014. doi: 10.4171/CMH/342