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On the residue class distribution of the number of prime divisors of an integer. (English) Zbl 1257.11087
Let \(\Omega(n)\) denote the number of prime divisors of \(n\) counting multiplicity. T. Kubota and M. Yoshida proved that for every fixed positive integer \(m\) and every \(j\in\{0,1,\dots, m-1\}\) it holds that \[ \#\{ n\leq x, \Omega(n)\equiv j\pmod m\}=\frac{x}{m}+o(x) \] as \(x\) tends to infinity [Nagoya Math. J. 163, 1–11 (2001; Zbl 0986.11066)].
In the present paper, the authors derive the same result using Hall’s asymptotic estimate for the mean value of a multiplicative arithmetic function [R. R. Hall, Mathematika, 42, No. 1, 144–157 (1995; Zbl 0831.11050)].
In addition, building on the results of Kubota’s and Yoshida’s work, the present authors show that for each fixed integer \(m\geq 3\) and every \(j\in\{0,1,\dots, m-1\}\) the error term in the above equality can not be replaced by \(o(x^\alpha)\) for any \(\alpha<1\).

MSC:
11N37 Asymptotic results on arithmetic functions
11N60 Distribution functions associated with additive and positive multiplicative functions
11N25 Distribution of integers with specified multiplicative constraints
11M41 Other Dirichlet series and zeta functions
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References:
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