##
**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) = **sum**_{p|n, p\alpha} \leq n < p^{\alpha+1}p^{\alpha}. Among other things, the author shows that m(x) = **max**_{n \leq x}f(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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag