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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 022.01001

**Autor: ** Erdös, Paul

**Title: ** On the smoothness of the asymptotic distribution of additive arithmetical functions. (In English)

**Source: ** Amer. J. Math. 61, 722-725 (1939).

**Review: ** Let f be an additive function of the form f_{n} = **sum** a_{p}, where the sum is over the prime divisors of n and a_{2}, a_{3},a_{4},... are real. If this f has an asymptotic distribution function \sigma then \sigma is known to be either purely discontinuous or purely singular or absolutely continuous. Conditions for the first case are well known. Examples are given for the other cases. In particular examples for which \sigma has derivatives of arbitrarily high order, and examples for which \sigma is represented by the values on the real axis of a transcendental entire function. The paragraph: "In fast,..." on p. 725 is damaged by misprints, but remains understandable.

**Reviewer: ** v.Kampen (Baltimore.)

**Classif.: ** * 11N60 Distribution functions (additive and positive multipl. functions)

11K65 Arithmetic functions (probabilistic number theory)

**Index Words: ** Number theory

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