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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