scholarly journals Variance of sums in arithmetic progressions of divisor functions associated with higher degree L-functions in 𝔽q[t]

2019 ◽  
Vol 16 (05) ◽  
pp. 1013-1030
Author(s):  
Edva Roditty-Gershon ◽  
Chris Hall ◽  
Jonathan P. Keating
Keyword(s):  
Elliptic Curves ◽  
Conjugacy Classes ◽  
Arithmetic Progressions ◽  
Divisor Function ◽  
Legendre Curve ◽  
Divisor Functions ◽  
Matrix Integrals ◽  
Usual Formula ◽  
Special Case

We compute the variances of sums in arithmetic progressions of generalized [Formula: see text]-divisor functions related to certain [Formula: see text]-functions in [Formula: see text], in the limit as [Formula: see text]. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when [Formula: see text], in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual [Formula: see text]-divisor function, when the [Formula: see text]-function in question has degree one. They illustrate the role played by the degree of the [Formula: see text]-functions; in particular, we find qualitatively new behavior when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over [Formula: see text], and we illustrate them by examining in some detail the generalized [Formula: see text]-divisor functions associated with the Legendre curve.

scholarly journals A lower bound for the variance of generalized divisor functions in arithmetic progressions

2021 ◽  
Author(s):  
Daniele Mastrostefano
Keyword(s):  
Lower Bound ◽  
Large Class ◽  
Complex Number ◽  
Arithmetic Progressions ◽  
Divisor Function ◽  
Multiplicative Functions ◽  
Divisor Functions

AbstractWe prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a result for any $$\alpha $$ α -fold divisor function, for any complex number $$\alpha \not \in \{1\}\cup -\mathbb {N}$$ α ∉ { 1 } ∪ - N , even when considering a sequence of parameters $$\alpha $$ α close in a proper way to 1. Our work builds on that of Harper and Soundararajan, who handled the particular case of k-fold divisor functions $$d_k(n)$$ d k ( n ) , with $$k\in \mathbb {N}_{\ge 2}$$ k ∈ N ≥ 2 .

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

scholarly journals THE AVERAGE NUMBER OF SUBGROUPS OF ELLIPTIC CURVES OVER FINITE FIELDS

2020 ◽  
Vol 71 (3) ◽  
pp. 781-822
Author(s):  
Corentin Perret-Gentil
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Short Interval ◽  
Divisor Function ◽  
Finite Abelian Groups ◽  
Cyclic Subgroups ◽  
Number Of Divisors ◽  
Curves Over Finite Fields ◽  
Euler Products

Abstract By adapting the technique of David, Koukoulopoulos and Smith for computing sums of Euler products, and using their interpretation of results of Schoof à la Gekeler, we determine the average number of subgroups (or cyclic subgroups) of an elliptic curve over a fixed finite field of prime size. This is in line with previous works computing the average number of (cyclic) subgroups of finite abelian groups of rank at most $2$. A required input is a good estimate for the divisor function in both short interval and arithmetic progressions, that we obtain by combining ideas of Ivić–Zhai and Blomer. With the same tools, an asymptotic for the average of the number of divisors of the number of rational points could also be given.

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 INFINITUDE OF ELLIPTIC CARMICHAEL NUMBERS

2012 ◽  
Vol 92 (1) ◽  
pp. 45-60 ◽  
Author(s):  
AARON EKSTROM ◽  
CARL POMERANCE ◽  
DINESH S. THAKUR
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Residue Class ◽  
Complex Multiplication ◽  
Weak Form ◽  
Arithmetic Progressions ◽  
Carmichael Numbers ◽  
The Third ◽  
Fermat’S Little Theorem

AbstractIn 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

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