## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  448.10040
Autor:  Erdös, Paul; Sarközy, A.
Title:  On the number of prime factors of integers. (In English)
Source:  Acta Sci. Math. 42, 237-246 (1980).
Review:  Let pii(x) be the number of integers n \leq x such that \Omega(n) = i, where \Omega(n) denotes the number of prime factors of n counted with multiplicity. Let \delta be a constant satisfying 0 < \delta < 2. Then the authors prove the following two results. First 2ii-4\pii(x) = 0(x log x) uniformly for all i \geq 1. Next (i-1)!(log log x)1-i = 0(\frac x{log x}) uniformly for all i satisfying 1 \leq i \leq (2-\delta) log log x. They deduce some corollaries to these results. We may quote: for every \epsilon > 0

sum1 \leq i \leq z log log k\pii(k) = 0(k(log k)-\phi(z)+\epsilon)   and sum1 \leq i \leq z log log k\pii(k2) = 0(k2(log k)-\phi(z)+\epsilon).

Here the 0-constant depends only on \epsilon and z. \phi(x) = 1*x log x-x, is defined for all x > 0 and z is defined as the unique real root of \phi(x+1) = \phi(x). It may be noted that z = 0.54....
Reviewer:  K.Ramachandra
Classif.:  * 11N37 Asymptotic results on arithmetic functions

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