Generalized divisor functions in arithmetic progressions: I

2021 ◽  
Author(s):  
David T. Nguyen
Keyword(s):  
Arithmetic Progressions ◽  
Divisor Functions

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 General divisor functions in arithmetic progressions to large moduli

2016 ◽  
Vol 59 (9) ◽  
pp. 1663-1668 ◽  
Author(s):  
Fei Wei ◽  
BoQing Xue ◽  
YiTang Zhang
Keyword(s):  
Arithmetic Progressions ◽  
Divisor Functions

scholarly journals Colorings with only rainbow arithmetic progressions

2020 ◽  
Vol 161 (2) ◽  
pp. 507-515
Author(s):  
J. Pach ◽  
I. Tomon
Keyword(s):  
Arithmetic Progressions

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

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