On the distribution of totients (Q2785031)

An integer \(n\) is called a totient (nontotient) if the equation \(\varphi(x)=n\) has at least one solution \(x\) (has no solutions in \(x\)), where \(\varphi\) is Euler's arithmetical function. NEWLINENEWLINENEWLINEIt is shown that there exists an absolute constant \(c\) such that for every integer \(d\) one has \(\# \{p\leq x: p\) is prime and \(dp\) is a totient\(\} \leq c \tau(d^2) \frac{x}{\log^2 x}\), where \(\tau(m)\) is the number of divisors of \(m\). A sufficient condition for nontotients of the form \(dp\) is also proved.