Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  329.10036
Autor:  Erdös, Paul; Hall, R.R.
Title:  Distinct values of Euler's \phi-function. (In English)
Source:  Mathematika, London 23, 1-3 (1976).
Review:  Let V(x) denote the number of distinct values not exceeding x taken by Euler's \phi-function, so that \pi (x) \leq V(x) \leq x. In a previous paper by the authors [Acta arithmetica 22, 201-206 (1973; Zbl 252.10007)], they show that for each fixed B > 2 \sqrt{(2/ log 2)}, the estimate V(x) << \pi (x) \exp {B \sqrt{(log log x)} } holds. In this paper they show that there exist positive absolute constants A,C, such that

V(x) \geq C   \pi (x) \exp {A(log log log x)2 }.

The methods used involve the number of representations of n in the form n = mi(p-1) where p is a prime and the sequence of distinct numbers of the form (p1-1)(p2-1) ... (pk-1) subject to various conditions. The authors conclude by asking the question: is it true that, for every c > 1, lim V(cx)/V(x) = c?
Reviewer:  E.M.Horadam
Classif.:  * 11N37 Asymptotic results on arithmetic functions
                   11A25 Arithmetic functions, etc.

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