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 Chebyshevskii Sb., 2019, Volume 20, Issue 2, Pages 123–139 (Mi cheb757)

Distribution of values of Jordan function in residue classes

L. A. Gromakovskaya, B. M. Shirokov

Petrozavodsk State University (Petrozavodsk)

Abstract: The concept of a uniform distribution of integral-valued arithmetic functions in residue classes modulo $N$ was introduced by I. Niven [3]. For multiplicative functions, the concept of a weakly uniform distribution modulo $N$, which was introduced by V. Narkevich [6], turned out to be more convenient. In papers on the distribution in residue classes, we usually give asymptotic formulas for the number of hits of the values of functions in a particular class containing only the leading terms, which is explained by the application to the generating functions of the Tauberian theorem of H. Delange [12], although these generating functions have better analytical properties, which is necessary for the theorem of H. Delange. In this paper we consider the distribution of values of the Jordan function $J_2(n)$. For a positive integer $n$, the value of $J_2(n)$ is the number of pairwise incongruent pairs of integers that are primitive in modulo $n$. It is proved that $J_2(n)$ is weakly uniformly distributed modulo $N$ if and only if $N$ is relatively prime to $6$. Moreover, the paper contains an asymptotic formula representing an asymptotic series, which is achieved by applying Lemma 3, which is a Tauberian theorem type that replaces the theorem of H. Delange.

Keywords: tauberian theorem, distribution of values, residue classes.

DOI: https://doi.org/10.22405/2226-8383-2018-20-2-123-139

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UDC: 511.3
Received: 07.12.2017
Accepted:12.07.2019

Citation: L. A. Gromakovskaya, B. M. Shirokov, “Distribution of values of Jordan function in residue classes”, Chebyshevskii Sb., 20:2 (2019), 123–139

Citation in format AMSBIB
\Bibitem{GroShi19} \by L.~A.~Gromakovskaya, B.~M.~Shirokov \paper Distribution of values of Jordan function in residue classes \jour Chebyshevskii Sb. \yr 2019 \vol 20 \issue 2 \pages 123--139 \mathnet{http://mi.mathnet.ru/cheb757} \crossref{https://doi.org/10.22405/2226-8383-2018-20-2-123-139} 

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