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Distribution of values of $$sigma_2(n)$$ in residue classes. (English) Zbl 0487.10037
An arithmetical integer-valued function $$f(n)$$ is called weakly uniformly distributed $$\pmod N$$ $$(\text{WUD}\pmod N)$$ if those of its values which are prime to $$N$$ are uniformly distributed among the restricted residue classes mod $$N$$. The set of integers $$N$$ for which $$f$$ is $$\text{WUD}\pmod N$$ is denoted by $$M(f)$$. In the present paper the set $$M(f)$$ for $$f=\sigma_2(n) = \sum_{d\mid n}d^2$$ is determined. The main result is the following proposition.
Theorem. The function $$\sigma_2(n)$$ is $$\text{WUD}\pmod N$$ for all integers $$N$$, except when (i) $$8\mid N$$, $$40\nmid N$$, or (ii) $$40\mid N$$ and $$N$$ has a prime divisor $$p\geq 1$$ such that the order of $$4\pmod p$$ is odd, or finally (iii) $$N$$ is a multiple of $$12, 15, 28, 42$$ or $$66$$.
The proof of this theorem is based on five lemmas. Lemma 1 had already been proved in [the first author, Acta Arith. 12, 269–279 (1967; Zbl 0147.29802)] and lemma 2 in [the first author, J. Reine Angew. Math. 323, 200–212 (1973; Zbl 0447.10050)].
Reviewer: L. Kuipers

##### MSC:
 11N64 Other results on the distribution of values or the characterization of arithmetic functions 11N37 Asymptotic results on arithmetic functions 11J71 Distribution modulo one
##### Citations:
Zbl 0147.29802; Zbl 0447.10050
Full Text:
##### References:
 [1] Narkiewicz, W.: On distribution of values of arithmetical functions in residue classes. Acta Arithm.12, 269-279 (1967). · Zbl 0147.29802 [2] Narkiewicz, W.: On a kind of uniform distribution for systems of multiplicative functions. Litovskij Mat. Sbornik22, 135-145 (1982). · Zbl 0496.10028 [3] Narkiewicz, W.: Euler’s ?-function and the sum of divisors. J. reine angew. Math.323, 200-212 (1981). · Zbl 0447.10050 [4] ?liwa, J.: On distribution of values of ? (n) in residue classes. Coll. Math.27, 283-291 (1973). · Zbl 0263.10016
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