Publications of (and about) Paul Erdös
Autor: Erdös, Pál; Schinzel, A.
Title: Distributions of the values of some arithmetical functions (In English)
Source: Acta Arith. 6, 473-485 (1961).
Review: A.Schinzel und Y. Wang proved: Given a1,a2,a3,...,ah \geq 0, \epsilon > 0, there exist c > 0 and x0 > 0 such that the number of positive integers n \leq x with |\phi(n+i)/\phi(n+i-1)-ai| < \epsilon(1 \leq i \leq h) is greater than cx/ logh+1x if x > x0 (Zbl 070.04201; Zbl 081.04203). Here \phi is Euler's function. A similar result was proven for \sigma. Shao Pin Tsung (Zbl 072.03304) extended these results to all positive multiplicative functions satisfying certain density conditions. The present paper strengthens the results by way of replacing the lower estimate cx/ logh+1 x by cx for positive multiplicative functions fs satisfying: sum (fs(p)-ps)2 p-2s-1 < oo (over primes p) and there exists an interval < a,b>, with a = 0 or b = oo, such that for every integer M > 0 the set of numbers fs(N)/Ns, with (N,M) = 1, is dense in < a,b>.
Classif.: * 11N64 Characterization of arithmetic functions
Index Words: number theory
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