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On the values of Euler’s \(\varphi\)-function. (English) Zbl 0252.10007
Let \(M\) denote the set of distinct values of Euler’s \(\varphi\)-function, let \(m_1,m_2,m_3, \ldots\) be the elements of \(M\) arranged as an increasing sequence and let \(V(x)= \sum_{m_i \leq x}1\). 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 \(m_{i+1}-m_i= \omega (\log m_i)\). 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 \[ (1+y)^{\omega'(n)}= \sum'_{d \mid n} y^{\nu (d)}(1+y)^{\omega (d)- \nu (d)} \] 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 \(\sum_{\omega \{\varphi (m) \} <2 \log \log x/\log 2}(1/m)\) is found using complex variable methods.
Reviewer: E.M.Horadam

11A25 Arithmetic functions; related numbers; inversion formulas
11N37 Asymptotic results on arithmetic functions
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