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

2014 ◽  
Vol 89 (4) ◽  
pp. 979-1014 ◽  
Author(s):  
Étienne Fouvry ◽  
Satadal Ganguly ◽  
Emmanuel Kowalski ◽  
Philippe Michel
Keyword(s):  
Gaussian Distribution ◽  
Arithmetic Progressions ◽  
Divisor Function ◽  
Hecke Eigenvalues

Fourier Coefficients of Modular Forms of Half-Integral Weight in Arithmetic Progressions

2020 ◽  
Author(s):  
Corentin Darreye
Keyword(s):  
Gaussian Distribution ◽  
Modular Forms ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Mixed Distribution ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

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.

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

2012 ◽  
Vol 153 (3) ◽  
pp. 419-455 ◽  
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.

scholarly journals The variance of divisor sums in arithmetic progressions

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.

scholarly journals The divisor function over arithmetic progressions

1992 ◽  
Vol 61 (3) ◽  
pp. 271-287 ◽  
Author(s):  
Etienne Fouvry ◽  
Henryk Iwaniec ◽  
Nicholas Katz
Keyword(s):  
Arithmetic Progressions ◽  
Divisor Function
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