Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 018.29301
Autor: Erdös, Paul
Title: On the density of some sequences of numbers. III. (In English)
Source: J. London Math. Soc. 13, 119-127 (1938).
Review: The author extends his previous work (see Zbl 012.01004 and Zbl 016.01204) on the distribution of the values of an additive arithmetical function f(m). The restriction f(m) \geq 0 is removed, and the results obtained is the present paper include those proved by I.J.Schoenberg (see Zbl 013.39302) using analytical methods. The main results are:
(1) If sump, |f(p)| > 1 1/p , sump, {|f(p)| \leq 1} {f(p) \over p}, sump, {|f(p)| \leq 1} {f2(p) \over p} (p running through primes) all converge, then the distribution-function for f(m) exists.
(2) If sumf(p) \ne 0 1/p diverges, the distribution-function is continuous, and if it converges, the distribution-function is purely discontinuous. The proofs are elementary, but more complicated than those of I and II.
Reviewer: Davenport (Manchester)
Classif.: * 11N60 Distribution functions (additive and positive multipl. functions)
Index Words: Number theory
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