Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 593.10036
Autor: Erdös, Paul
Title: On two unconventional number-theoretic functions and on some related problems. (In English)
Source: Calcutta Math. Soc. Diamond-Cum-Platinum Jubilee Commem. Vol. (1908- 1983), Pt. 1, 113-121 (1984).
Review: [For the entire collection see Zbl 584.00012.]
The author proves a number of results and formulates conjectures about two number-theoretic functions related to the distribution of the prime divisors of an integer. One of the two functions is defined as f(n) = sump|n, p\alpha \leq n < p^{\alpha+1}p\alpha. Among other things, the author shows that m(x) = maxn \leq xf(n) satisfies
m(x) \leq (1+o(1))x log x/ log log x as x ––> oo, and conjectures that in this bound one has asymptotic equality. He further states that the logarithmic density of the set of integers n satisfying f(n) \leq cn exists for any c and is a continuous function of c.
Reviewer: A.Hildebrand
Classif.: * 11N37 Asymptotic results on arithmetic functions
11K65 Arithmetic functions (probabilistic number theory)
Keywords: arithmetic functions; conjectures; prime divisors; logarithmic density
Citations: Zbl 584.00012
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