Distribution of values of \(sigma_2(n)\) in residue classes.

*(English)*Zbl 0487.10037An 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)].

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 |

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\textit{W. Narkiewicz} and \textit{F. Rayner}, Monatsh. Math. 94, 133--141 (1982; Zbl 0487.10037)

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##### 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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