Répartition des valeurs de la fonction d'Euler. (Distribution of values of Euler's function) (Q1825220)

Denote by \(\phi\) Euler's function and let F(x) be the number of values of the \(\phi\)-function less than or equal to x. The best known result for the error term R(x) defined by \[ F(x)=\zeta (2)\zeta (3)/\zeta (6)\cdot x+R(x) \] is due to \textit{P. T. Bateman} [Acta Arith. 21, 329-345 (1972; Zbl 0217.319)]. Utilizing methods of complex analysis he showed \[ R(x)\quad \ll \quad x\cdot \exp (-(1-\epsilon)( \log x \log \log x)^{1/2}) \] for any \(\epsilon >0\). Using only elementary means \textit{J. L. Nicolas} [Enseign. Math., II. Ser. 30, 331-338 (1984; Zbl 0553.10036)] proved R(x) \(\ll x/\log x\). Generalizing this method the author improves the result to R(x) \(\ll x/\log^ 2 x.\) Reviewer's remark. The author conjectured that elementary methods lead to an estimation R(x) \(\ll x/\log^ k x\) for any \(k>0\). In the meantime \textit{M. Balazard} and the author have even proved \[ R(x)\quad \ll \quad x\cdot \exp (-c\sqrt{\log x}) \] which is very close to the above mentioned result of P. T. Bateman.