Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 252.10007
Autor: Erdös, Paul; Hall, R.R.
Title: On the values of Euler's \phi-function. (In English)
Source: Acta Arith. 22, 201-206 (1973).
Review: Let M denote the set of distinct values of Euler's \phi-function, let m1,m2,m3, ... be the elements of M arranged as an increasing sequence and let V(x) = summi \leq x1. The authors prove the main result that for each B > 2 \sqrt{2/ log 2}, V(x) = 0(\pi (x) \exp {B \sqrt{log log x}}) and conjecture that mi+1-mi = \omega (log mi). Let \omega (n) denote the number of prime factors of n counted according to multiplicity, \omega'(n) the number of odd prime factors of n and \nu (n) the number of distinct prime factors of n. By considering the identity
(1+y)\omega'(n) = sum'd | n y\nu (d)(1+y)\omega (d)- \nu (d) where sum' denotes a sum restricted to odd d it is shown that the number of integers n \leq x for which \omega (n) \geq 2 log log x/ log 2 is 0(\pi (x) log log x). From this the main result is proved by dividing the integers n \leq x into two special classes and by dividing V(x) into two sums over different subsets of M. An auxiliary result evaluating sum\omega {\phi (m) < 2 log log x/ log 2}(1/m) is found using complex variable methods.
Reviewer: E.M.Horadam
Classif.: * 11A25 Arithmetic functions, etc.
11N37 Asymptotic results on arithmetic functions
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