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On sums involving reciprocals of certain large additive functions. (English) Zbl 0682.10030

For \(n\geq 2\) let P(n) denote the largest prime factor of n, and let \(\omega (n)=\sum_{p | n}1\), \(\Omega (n)=\sum_{p^{\alpha} \| n}\alpha\), \(\beta (n)=\sum_{p | n}p\), \(B(n)=\sum_{p^{\alpha} \| n}\alpha p\), \(B_ 1(n)=\sum_{p^{\alpha} \| n}p^{\alpha}\). The reviewer [Arch. Math. 36, 57-61 (1980; Zbl 0436.10019)] proved \((\log_ kx=\log (\log_{k-1}x))\) \[ \sum_{2\leq n\leq x}1/P(n)=x \exp \{-(2 \log x \log_ 2x)^{1/2}+O((\log x \log_ 3x)^{1/2})\}, \] and C. Pomerance and the reviewer [Topics in classical number theory, Coll. Math. Soc. J. Bolyai 34, 769-789 (1984; Zbl 0546.10037)] improved and generalized this result. An explicit asymptotic formula for the above sum (and estimates for various sums involving P(n) and the additive functions defined above) was obtained by P. Erdős, A. Ivić and C. Pomerance [Glas. Mat., III. Ser. 21(41), 283-300 (1986; Zbl 0615.10055)] and the reviewer [Acta Arith. 49, 21-32 (1987; Zbl 0573.10038)].
The present author builds on the methods used in the papers quoted above and employs a sophisticated elementary technique to obtain precise relations between several sums. He proves \((1)\quad \sum_{2\leq n\leq x}\frac{1}{\beta (n)}=(D+O(\log^ 2_ 3x/\log_ 2x))\sum_{2\leq n\leq x}\frac{1}{P(n)},(2)\sum_{2\leq n\leq x}\frac{\omega (n)}{\beta (n)}=D(\frac{2 \log x}{\log_ 2x})^{1/2}(1+O(\log^ 2_ 3x/\log_ 2x))\sum_{2\leq n\leq x}\frac{1}{P(n)},(3)\quad \sum_{2\leq n\leq x}\frac{\Omega (n)-\omega (n)}{\beta (n)}=(D\sum_{p}\frac{1}{p^ 2- p}+O(\log^ 2_ 3/\log_ 2x))\sum_{2\leq n\leq x}\frac{1}{P(n)}.\)Here \(<D<1\) is an absolute constant that is precisely defined in the text. The formulas (1)-(3) remain valid if \(\beta\) (n) is replaced by B(n) or \(B_ 1(n)\) and (2) remains valid if \(\omega\) (n) is replaced by \(\Omega\) (n). The author’s results are remarkably sharp, and elucidate the relation between “small” additive functions \(\omega\) (n), \(\Omega\) (n) and reciprocals of “large” additive functions \(\beta\) (n), B(n) and \(B_ 1(n)\). A systematic account of reciprocals of arithmetical functions is given in the monograph of J.-M. De Koninck and the reviewer [Topics in arithmetical functions (North Holland 1980; Zbl 0442.10032)].
Reviewer: A.Ivić

MSC:

11N37 Asymptotic results on arithmetic functions
11N05 Distribution of primes
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