Distribution of values of Jordan function in residue classes

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.