A concentration property with respect to the Euler function
Let \(E(n)=\Omega (\varphi (n))\) and \[ Q(x,E) = x^{-1} \max _k \# \{n\leq x: E(n)=k \} . \] \textit{P. Erdős} and \textit{C. Pomerance} [Rocky Mt. J. Math. 15, 343-352 (1985; Zbl 0617.10037)] proved that \(E\) has a normal limiting distribution with norming \( ( \log\log x)^{3/2} \). This led \textit{M. Balazard} and \textit{A. Smáti} [Groupe de Travail en Théorie Analytique et Élémentaire des Nombres 1989-1990, Publ. Math. Orsay 92-01, 5-36 (1992; Zbl 0780.11005)] to conjecture that \(Q(x, E) \asymp (\log\log x)^{-3/2}\). This conjecture is established in the present paper. This requires rather elaborate study of this function since (1) the number-theoretical analogs of the Kolmorov-Rogozin inequality, as given by Halász and the reviewer, are principally uncapable to give a better estimate than \((\log\log x)^{-1/2}\), and (2) as the value of \(E(p)\) depends on multiplicative properties of \(p-1\), even the simple auxiliary properties of \(E\) are often difficult to establish.
scientific article; zbMATH DE number 1251200
scientific article