Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  422.10035
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:  Sémin. Delange-Pisot-Poitou, 20e Année 1978/79, Théorie des nombres, Fasc. 2, Exp. 32, 19 p. (1980).
Review:  [For the entire collection see Zbl 418.00004.]
The authors obtain significant results concerning \omega(n) and \Omega(n), which represent the number of distinct prime factors of n and the number of total prime factor of n respectively. Defining \omega-largely composite numbers as those n for which m \leq n implies \omega(m) \leq \omega(n), they first prove that for some constants 0 < c1 < c2

\exp(c1 log ½x) \leq Q\ell(x) \leq \exp(c2 log ½x)

where Q\ell(x) is the number of \omega-largely composite numbers not exceeding x. The Brum-Titchmarsh inequality and A.Selberg's result on primes in short intervals [Arch. Math. Naturvid. B 47, No. 6, 1-19 (1943; Zbl 028.34802)] are needed in the proof, and reasons for conjecturing that

log Q\ell(x) = (1+o(1))\pi(2/3) ½ log ½x,  x ––> oo

are stated. Next an asymptotic formula for fc(x), the number of n \leq x for which \omega(n)c log x/ log log x, is derived, where 0 < c < 1 is given. The result is fc(x) = x1-c+o(1), the proof being elementary and elegant. The restriction c < 1 is natural, since \omega(n) \leq (1+o(1)) log n/ log log n as n ––> oo. Properties of \omega-interesting numbers (defined as n > 1 which satisfy \omega(m)/m < \omega(n)/n for m > n) are extensively discussed, and maximal and minimal order of f(n)+f(n+1) is investigated when f(n) is \sigma(n), \phi(n) and \Omega(n). In the last case it is proved that, as n ––> oo,

\Omega(n)+\Omega(n+1) \leq (1+o(1)) log n/ log 2,

which is best possible, since \Omega(n) \leq log n/ log 2 is already attained when n is a power of two. The corresponding problem for \omega(n), i.e. determining

lim\supn ––> oo(\omega(n)+\omega(n+1))\frac{log log n}{log n},

remains yet to be solved.
Reviewer:  A.Ivic
Classif.:  * 11N37 Asymptotic results on arithmetic functions
                   11N05 Distribution of primes
Keywords:  number of prime factors of integers; extremal values; omega-largely composite numbers; omega-interesting numbers
Citations:  Zbl.028.348; Zbl.418.00004

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