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**Zentralblatt MATH**

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

**Zbl.No: ** 104.27202

**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 a_{1},a_{2},a_{3},...,a_{h} \geq 0, \epsilon > 0, there exist c > 0 and x_{0} > 0 such that the number of positive integers n \leq x with |\phi(n+i)/\phi(n+i-1)-a_{i}| < \epsilon(1 \leq i \leq h) is greater than cx/ log^{h+1}x if x > x_{0} (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/ log^{h+1} x by cx for positive multiplicative functions f_{s} satisfying: **sum** (f_{s}(p)-p^{s})^{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 f_{s}(N)/N^{s}, with (N,M) = 1, is dense in < a,b>.

**Reviewer: ** J.P.Tull

**Classif.: ** * 11N64 Characterization of arithmetic functions

**Index Words: ** number theory

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