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

Abstract We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry–Ganguly–Kowalski–Michel and Kowalski–Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution that comes from the asymptotic mixed distribution of Salié sums.

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Manin's conjecture for a cubic surface with 2A2 + A1 singularity type

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500411200031x ◽

2012 ◽

Vol 153 (3) ◽

pp. 419-455 ◽

Cited By ~ 5

Author(s):

PIERRE LE BOUDEC

Keyword(s):

Arithmetic Progressions ◽

Divisor Function ◽

Cubic Surface ◽

Singularity Type ◽

Manin’S Conjecture ◽

A1 Singularity

AbstractWe establish Manin's conjecture for a cubic surface split over ℚ and whose singularity type is 2A2 + A1. For this, we make use of a deep result about the equidistribution of the values of a certain restricted divisor function in three variables in arithmetic progressions. This result is due to Friedlander and Iwaniec (and was later improved by Heath–Brown) and draws on the work of Deligne.

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THE DIVISOR FUNCTION IN ARITHMETIC PROGRESSIONS MODULO PRIME POWERS

Mathematika ◽

10.1112/s0025579316000024 ◽

2016 ◽

Vol 62 (3) ◽

pp. 898-908 ◽

Cited By ~ 4

Author(s):

Rizwanur Khan

Keyword(s):

Arithmetic Progressions ◽

Divisor Function

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The variance of divisor sums in arithmetic progressions

Forum Mathematicum ◽

10.1515/forum-2016-0227 ◽

2018 ◽

Vol 30 (2) ◽

pp. 269-293

Author(s):

Brad Rodgers ◽

Kannan Soundararajan

Keyword(s):

Arithmetic Progressions ◽

Large Sieve ◽

Restricted Range ◽

Divisor Function ◽

Residue Classes ◽

Divisor Sums

AbstractWe study the variance of sums of thek-fold divisor function{d_{k}(n)}over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture about the asymptotics of this variance. This result is closely related to moments of DirichletL-functions, and our proof relies on the asymptotic large sieve.

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