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**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 **sum**_{p, |f(p)| > 1} ^{1}/_{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) \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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