×

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
PDFBibTeX XMLCite
Full Text: DOI EuDML

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.