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 fk(n) = max{g(n+j): j = 1,...,k}. The authors give conditions which imply that for every \epsilon > 0 and every k0 the inequality fk(n) < (1+\epsilon)fk(0) holds for k \geq k0 and all but \delta(\epsilon,k0)x integers n in [1,x], \delta(\epsilon,k0) ––> 0 for k0 ––> 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

0k(n) = max{\omega(n+j): j = 1,...,k},ok(n) = max{\omega(n+j): j = 1,...,k}.

The authors prove, by use of Brun's sieve, that for every \epsilon > 0 the inequalities ( log2 = log log)

0k(n) \geq (1-\epsilon)\rho(\frac{log k}{log2n}) log2n, ok(n) \leq (\overline{\rho}(\frac{log k}{log2n})+\epsilon) log2n

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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