
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 integralvalued 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/10.22405/222683832018202123139
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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 123139
\mathnet{http://mi.mathnet.ru/cheb757}
\crossref{https://doi.org/10.22405/222683832018202123139}
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