A measure of the nonmonotonicity of the Euler phi function (Q1235733)

Let \(f\) be a real valued arithmetic function satisfying \(\lim_{n\to\infty} f(n) = +\infty\). Define another arithmetic functions \(F = F_f\) by setting \[ F_f(n) = \# \{j < n: f(j) \geq f(n)\} + \# \{j> n:f(j) \leq f(n)\}. \] The size of the values assumed by the function F provides a measure of the nonmonotonicity of f. In particular, F is identically zero if an only if f is strictly increasing. In the present article \(f = \varphi\) , Euler's functions and \(F_{\varphi}\) is written as \(F\). It is shown that \(F(n)/n\) is asymptotically represented as \(h(\varphi(n)/n)\), where \(h\) is a certain convex function. Using this representation it is shown that \(F(n)/n\) has a distribution function. The functions \(\max_{n \leq x} F(n)\) and \(\min_{n > x} F(n)\) are studied and conditions on \(\varphi(n)/n\) are found which lead to large and small values of \(F(n)/n\).