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 $\left(d,\frac nd\right)=1$. The sum of unitary divisors of a number $n$ is denoted by $\sigma^*(n)$. For a fixed function $f(n)$ let us denote by $S(x,r)$ the numbers of positive integers $n\le x$ such that $f(n)\equiv r\mod N$ 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 $a$, $b$
$$ \lim_{x\to\infty}\frac{S(x,a)}{S(x,b)}=1 $$
provided that the set $\{r\mid(r,N)=1\}$ is infinite. We use the tauberian theorem to obtain an asymptotic series for $S(x,r)$ for $\sigma^*(n)$. Then we derive necessary and sufficient conditions for the module $N$ that provide weakly uniform distribution modulo $N$ of the function $\sigma^*(n)$.