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 Pi(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 Pi(n) is the i-th largest prime factor of n, i.e. P(n) = P1(n) > P2(n) > ... > Pi(n) > ... > P\omega(n)(n) are the distinct prime factors of n. It is shown that the sums sum2 \leq n \leq xn-1/\omega(n) and sum2 \leq n \leq xn-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) ½sum2 \leq n \leq x1/P(n) << sum2 \leq n \leq x\Omega(n)/P(n) << (log x log log x) ½sum2 \leq n \leq x1/P(n), (1) and in a joint forthcoming paper with C. Pomerance (1) will be further sharpened. Finally it is shown that with some Bi > 0 and i \geq 2 fixed
sum'n \leq x1/Pi(n) = (Bi+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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