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**Zentralblatt MATH**

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

**Zbl.No: ** 417.10039

**Autor: ** Erdös, Paul; Katai, I.

**Title: ** On the growth of some additive functions on small intervals. (In English)

**Source: ** Acta Math. Acad. Sci. Hung. 33, 345-359 (1979).

**Review: ** Let g: **N** ––> **R** denote a non-negative strongly additive function, and let f_{k}(n) = **max****{**g(n+j): j = 1,...,k**}**. The authors give conditions which imply that for every \epsilon > 0 and every k_{0} the inequality f_{k}(n) < (1+\epsilon)f_{k}(0) holds for k \geq k_{0} and all but \delta(\epsilon,k_{0})x integers n in [1,x], \delta(\epsilon,k_{0}) ––> 0 for k_{0} ––> oo. Some questions concerning the necessity of the conditions remain open. The main part of the paper is devoted to the special case g = \omega, where \omega(n) denotes the number of distinct prime factors of n in **N**. Let 0_{k}(n) = **max****{**\omega(n+j): j = 1,...,k**}**,o_{k}(n) = **max****{**\omega(n+j): j = 1,...,k**}**. The authors prove, by use of Brun's sieve, that for every \epsilon > 0 the inequalities ( log_{2} = log log)

0_{k}(n) \geq (1-\epsilon)\rho**(**\frac{log k}{log_{2}n}**)** log_{2}n, o_{k}(n) \leq (\overline{\rho}**(**\frac{log k}{log_{2}n}**)**+\epsilon) log_{2}n hold for every k \geq 1 apart from a set of n's having zero density. Here \rho, \overline{\rho} are defined as the inverse functions of \Psi with \Psi(r) = r log ^{r}/_{e} +1 for r \geq 1 resp. 0 < r \leq 1, \overline{\rho}(u) = 0 for u \geq 1. This result corresponds to similar upper resp. lower bounds obtained by *I.Kátai* [Publ. Math., Debrecen 18, 171-175 (1971; Zbl 261.10029)].

**Reviewer: ** L.Lucht

**Classif.: ** * 11N35 Sieves

11N05 Distribution of primes

11N37 Asymptotic results on arithmetic functions

**Keywords: ** sieve methods; additive functions; growth; strongly additive function; number of distinct prime factors

**Citations: ** Zbl.261.10029

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