for two different primes p₁,p₂; and let

The main result of this paper, which constitutes a wide generalization of the author's work on abundant numbers (Zbl 010.10303), is that lim_{n ––> oo} N (f; c)/n exists and is a continuous function of c. The case of the abundant numbers is obtained by taking f(m) = log {\sigma(m) \over m}, c = log 2.
Suppose first f(m) satisfies the more stringent conditions: (3) f(p^\alpha) = f(p), (4) sum_p {f(p)\over p} converges. Defining f_p(m) = sum_{p|m, {p \leq P}} f(p) it is easily seen that lim_{n ––> oo} N(f_p; c)/n = A_p exists, and as A_p in non-decreasing and \leq 1, lim_{p ––> oo} A_p = A exists. That lim_{n ––> oo} N(f; c)/n = A follows easily from the two lemmas:
(I) For any \epsilon > 0 there exists a \delta such that N(f; c,c+\delta) < \epsilon n for all sufficiently large n;
(II) For any \epsilon,\delta > 0 there exists a P(\epsilon,\delta) such that for P > P(\epsilon,\delta) and all n, the number of integers m \leq n for which f(m)-f_p(m) > \delta is less than \epsilon n. The main difficulty lies in the proof of (I), which uses the same idea as the paper already cited.
The author then sketches the proof when (3) is not assumed. As regards (4), he shows that it can be replaced by a weaker condition (4') and that if (4') does not hold, then lim_{n ––> oo} N(f; c)/n = 1.
Reviewer:  Davenport (Cambridge)
Classif.:  * 11N60 Distribution functions (additive and positive multipl. functions)
Index Words:  Number theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag

Books	Problems	Set Theory	Combinatorics	Extremal Probl/Ramsey Th.
Graph Theory	Add.Number Theory	Mult.Number Theory	Analysis	Geometry
Probabability	Personalia	About Paul Erdös	Publication Year	Home Page