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**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 = m_{i}(p-1) where p is a prime and the sequence of distinct numbers of the form (p_{1}-1)(p_{2}-1) ... (p_{k}-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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