Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  466.10037
Autor:  Erdös, Paul; Nicolas, Jean-Louis
Title:  Sur la fonction: Nombre de facteurs premiers de N. (On the function: Number of prime factors of N.) (In French)
Source:  Enseign. Math., II. Ser. 27, 3-27 (1981).
Review:  This paper considers several problems concerning the functions \omega(n) and \Omega(n). The following are proved: 1) Let Q1(x) be the number of n \leq x such that \omega(n) \leq \omega(m) whenever m \leq n. Then (log x) ½ << log Q1(x) << (log x)1/1. 2) For any fixed c > 0 has

\#\left{n \leq x; \omega(n) > \frac{c log x}{log log x}\right}x1-c+O(1).

3) lim\sup(log n)-1(\Omega(n)+\Omega(n+1)) = (log2)-1. 4) There exist infinitely many n for which m-\omega(m) < n-\omega(n) whenever m < n and m-\omega(m) > n-\omega(n) whenever m > n. 5) If \alpha > 1 is constant there is an asymptotic formula for \#{n \leq x; \omega(n) > \alpha log log x}, correct to within a factor 1-O((log log x)-1). The methods used are largely elementary, but an ineffective result on Diophantine approximation is also needed.
Reviewer:  D.R.Heath-Brown
Classif.:  * 11N37 Asymptotic results on arithmetic functions
                   11N05 Distribution of primes
Keywords:  number of prime factors; largely composite; total number of prime factors; asymptotic formula

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