DISTRIBUTION OF VALUES OF THE SUM OF UNITARY DIVISORS IN RESIDUE CLASSES

In this paper we prove the tauberian type theorem containing the asymptotic series for the Dirichlet series. We use this result to study distribution of sum of unitary divisors in residue classes coprime with a module. The divisor d of the integer n is an unitary divisor if (d, n/d) = 1. The sum...

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Main Authors: B. M. Shirokov, L. A. Gromakovskaya
Format: Article
Language:English
Published: Petrozavodsk State University 2016-05-01
Series:Проблемы анализа
Subjects:
Online Access:http://issuesofanalysis.petrsu.ru/article/genpdf.php?id=3370&lang=ru
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author B. M. Shirokov
L. A. Gromakovskaya
author_facet B. M. Shirokov
L. A. Gromakovskaya
author_sort B. M. Shirokov
collection DOAJ
description In this paper we prove the tauberian type theorem containing the asymptotic series for the Dirichlet series. We use this result to study distribution of sum of unitary divisors in residue classes coprime with a module. The divisor d of the integer n is an unitary divisor if (d, n/d) = 1. The sum of unitary divisors of a number n is denoted by σ∗(n). For a fixed function f(n) let us denote by S(x,r) the numbers of positive integers n ≤ x such that f(n) ≡ r(mod N) for x > 0 and r coprime with module N . According to W. Narkiewicz [5], a function f(n) is called weakly uniformly distributed modulo N if and only if for every pair of coprime integer a, b lim S(x,a)/ S(x,b)=1, (x→∞) provided that the set {r | (r,N) = 1} is infinite. We use the tauberian theorem to obtain an asymptotic series for S (x,r) for σ∗(n). Then we derive necessary and sufficient conditions for the module N that provide weakly uniform distribution modulo N of the function σ∗(n).
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spelling doaj.art-b5a2434cdc0349aaae4a0e35ac989cd22022-12-21T21:59:41ZengPetrozavodsk State UniversityПроблемы анализа2306-34242306-34322016-05-01523314410.15393/j3.art.2016.3370DISTRIBUTION OF VALUES OF THE SUM OF UNITARY DIVISORS IN RESIDUE CLASSESB. M. Shirokov0L. A. Gromakovskaya1 Petrozavodsk State University Petrozavodsk State UniversityIn this paper we prove the tauberian type theorem containing the asymptotic series for the Dirichlet series. We use this result to study distribution of sum of unitary divisors in residue classes coprime with a module. The divisor d of the integer n is an unitary divisor if (d, n/d) = 1. The sum of unitary divisors of a number n is denoted by σ∗(n). For a fixed function f(n) let us denote by S(x,r) the numbers of positive integers n ≤ x such that f(n) ≡ r(mod N) for x > 0 and r coprime with module N . According to W. Narkiewicz [5], a function f(n) is called weakly uniformly distributed modulo N if and only if for every pair of coprime integer a, b lim S(x,a)/ S(x,b)=1, (x→∞) provided that the set {r | (r,N) = 1} is infinite. We use the tauberian theorem to obtain an asymptotic series for S (x,r) for σ∗(n). Then we derive necessary and sufficient conditions for the module N that provide weakly uniform distribution modulo N of the function σ∗(n).http://issuesofanalysis.petrsu.ru/article/genpdf.php?id=3370&lang=rutauberian theoremsum of the unitary divisorsdistribution of values in the residue classes
spellingShingle B. M. Shirokov
L. A. Gromakovskaya
DISTRIBUTION OF VALUES OF THE SUM OF UNITARY DIVISORS IN RESIDUE CLASSES
Проблемы анализа
tauberian theorem
sum of the unitary divisors
distribution of values in the residue classes
title DISTRIBUTION OF VALUES OF THE SUM OF UNITARY DIVISORS IN RESIDUE CLASSES
title_full DISTRIBUTION OF VALUES OF THE SUM OF UNITARY DIVISORS IN RESIDUE CLASSES
title_fullStr DISTRIBUTION OF VALUES OF THE SUM OF UNITARY DIVISORS IN RESIDUE CLASSES
title_full_unstemmed DISTRIBUTION OF VALUES OF THE SUM OF UNITARY DIVISORS IN RESIDUE CLASSES
title_short DISTRIBUTION OF VALUES OF THE SUM OF UNITARY DIVISORS IN RESIDUE CLASSES
title_sort distribution of values of the sum of unitary divisors in residue classes
topic tauberian theorem
sum of the unitary divisors
distribution of values in the residue classes
url http://issuesofanalysis.petrsu.ru/article/genpdf.php?id=3370&lang=ru
work_keys_str_mv AT bmshirokov distributionofvaluesofthesumofunitarydivisorsinresidueclasses
AT lagromakovskaya distributionofvaluesofthesumofunitarydivisorsinresidueclasses