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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.