zbMATH — the first resource for mathematics

On the density of some sequences of numbers. III. (English) Zbl 0018.29301
The author extends his previous work (see Zbl 0012.01004 and Zbl 0016.01204) 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 0013.39302) using analytical methods. The main results are:
(1) If $$\sum_{p, |f(p)| > 1} {1\over p}$$, $$\sum_{p, {|f(p)| \leq 1}} {f(p) \over p}$$, $$\sum_{p, {|f(p)| \leq 1}} {f^2(p) \over p}$$ ($$p$$ running through primes) all converge, then the distribution-function for $$f(m)$$ exists.
(2) If $$\sum_{f(p) \neq 0} {1\over p}$$ 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.

MSC:
 11N60 Distribution functions associated with additive and positive multiplicative functions
Number theory
Full Text: