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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 506.10035

**Autor: ** Erdös, Paul; Ivic, Aleksandar

**Title: ** On sums involving reciprocals of certain arithmetical functions. (In English)

**Source: ** Publ. Inst. Math., Nouv. Ser. 32(46), 49-56 (1982).

**Review: ** The paper investigates certain sums involving reciprocals of the functions \omega(n), \Omega(n) and P_{i}(n). Here \omega(n) denotes the number of distinct prime factors of n, \Omega(n) denotes the number of all prime factors of n, and if \omega(n) \geq i then P_{i}(n) is the i-th largest prime factor of n, i.e. P(n) = P_{1}(n) > P_{2}(n) > ... > P_{i}(n) > ... > P_{\omega(n)}(n) are the distinct prime factors of n. It is shown that the sums **sum**_{2 \leq n \leq x}n^{-1/\omega(n)} and **sum**_{2 \leq n \leq x}n^{-1/\Omega(n)} are of the order x\exp(-A(log x· log log x)^{ ½}) and x\exp(-B(log x)^{ ½}) respectively, where the upper and lower bounds for A,B > 0 are explicitly calculable. Next it is shown that

(log x/ log log x)^{ ½}**sum**_{2 \leq n \leq x}1/P(n) << **sum**_{2 \leq n \leq x}\Omega(n)/P(n) << (log x log log x)^{ ½}**sum**_{2 \leq n \leq x}1/P(n), (1) and in a joint forthcoming paper with C. Pomerance (1) will be further sharpened. Finally it is shown that with some B_{i} > 0 and i \geq 2 fixed

**sum**'_{n \leq x}1/P_{i}(n) = (B_{i}+0(1))x(log log x)^{i-2}/ log x, where the dash' denotes summation over those n for which \omega(n) \geq i.

**Classif.: ** * 11N05 Distribution of primes

**Keywords: ** number of distinct prime factors; number of prime factors; largest prime factor

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