On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions \(\varphi\) and \(\sigma\) (Q2773362)

Let \(\varphi(n)\) and \(\sigma(n)\) denote, as usual, the Euler function and the sum of divisors of \(n\), respectively. \textit{A. Mąkowski} and \textit{A. Schinzel} [Colloq. Math. 13, 95-99 (1964; Zbl 0124.02702)] conjectured that \(S(n)\geq 1/2\) for all \(n\geq 1\), where \(S(n)=\sigma(\varphi(n))/n\). It is known that \(\limsup_n S(n)=\infty\) [see \textit{L. Alaoglu} and \textit{P. Erdős}, Bull. Am. Math. Soc. 50, 881-882 (1944; Zbl 0061.07808)] and in this paper it is asserted without proof that \(S(n)\to \infty\) on a set of asymptotic density \(1\). The authors of the present paper give a proof of this result in a stronger form. Namely, they prove the following theorems. NEWLINENEWLINENEWLINETheorem 1. (i) \(\limsup_n S(n)/\log_2 (n)=e^{\gamma}\). (ii) For each \(u\) with \(0\leq u \leq 1\), the asymptotic density of the set \(\{n: S(n)>ue^{\gamma} \log_3 (n)\}\) exists, it is a strictly decreasing function in \(u\) and varies continuously with \(u\). (iii) \(\frac 1{x} \sum_{n\leq x} S(n) =\frac{6e^{\gamma}}{\pi^2}\log_3 (x) +O((\log_3 (x))^{1/2})\). Here \(\log_k (x)\) stands for the \(k\)-fold iterated natural logarithm. NEWLINENEWLINENEWLINETheorem 2. Let \(\alpha =\liminf_n S(n)\). Then \(\{S(n): n\geq 1\}\) is dense in the interval \([\alpha, \infty]\). NEWLINENEWLINENEWLINEThey also prove, in a stronger form, a conjecture of P. Erdős stating that \(\varphi(n-\varphi(n))<\varphi(n)\) holds for almost all integers \(n>1\) (Theorem 3).