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**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 pi_{i}(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 2^{i}i^{-4}\pi_{i}(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 **sum**_{1 \leq i \leq z log log k}\pi_{i}(k) = 0(k(log k)^{-\phi(z)+\epsilon}) and **sum**_{1 \leq i \leq z log log k}\pi_{i}(k^{2}) = 0(k^{2}(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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