Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 802.11035
Autor: Erdös, Paul; Sárközy, A.
Title: On isolated, respectively consecutive large values of arithmetic functions. (In English)
Source: Acta Arith. 66, No.3, 269-295 (1994).
Review: This paper is divided into two parts. In the first part, the authors investigate the occurrence of isolated large values of the arithmetic functions \omega(n), \Omega(n), d(n), and \sigma(n). Given an arithmetic function f(n) and x \geq 1, the authors define G = G(f,x) as the largest integer such that the inequality f(n) > sum0 < |i| \leq G f(n+i) holds for some positive integer n \leq x, and obtain estimates for this quantity when f is one of the above functions. For example, in the case f = \omega the authors show that {{log x} \over {(log2 x)2}} << G(\omega, x) << {{log x} \over {log2 x log3 x}}, where logk x denotes the k fold iterated logarithm, and for f = \sigma they derive the asymptotic formula
G(\sigma, x) ~ {{3e\gamma} \over {\pi2}} log2 x (x ––> oo). The second part of the paper is devoted to the study of consecutive large values of the above arithmetic functions. Setting M(f,x) = maxn \leq x f(n), T(f,x) = maxn \leq x (f(n- 1)+f(n)), the authors show, for example, that
T(\Omega,x) \geq M(\Omega,x)+\exp { (log 2- \epsilon) {{log2 x} \over {log3 x}} } holds for any given \epsilon > 0 and arbitrarily large values \chi. Surprisingly, as the authors remark, the case of the function \omega(n) appears to be much more difficult, and even the weakest non-trivial estimate of this type, namely limsupx ––> oo (T(\omega,x)- M(\omega, x)) = oo, remains an open problem.
Reviewer: A.Hildebrand (Urbana)
Classif.: * 11N37 Asymptotic results on arithmetic functions
11N64 Characterization of arithmetic functions
11K65 Arithmetic functions (probabilistic number theory)
Keywords: occurrence of isolated large values; arithmetic functions; asymptotic formula; consecutive large values
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