An arithmetic function arising from Carmichael's conjecture (Q449718)

Let \(\varphi(n)\) denote Euler's function and \(F(n)= \#\{m: \varphi(n)\}\). Although Carmichael's conjecture that \(F(n)\geq 2\) for all \(n\geq 1\) remains unproven there are many other results concerning \(F(n)\) in the literature. For example, \textit{K. Ford} [Ann. Math. (2) 150, No. 1, 283--311 (1999; Zbl 0978.11053)] showed that every integer \(>1\) lies in the range of \(F\), and \textit{C. Pomerance} [Mathematika 27, 84--89 (1980; Zbl 0437.10001)] proved that \(\max_{n\leq x}\,F(n)\leq x/(L(x))^{1+o(1)}\) with \(L(x)= x^{\log\log\log x/\log\log x}\) and that under a certain plausible but difficult hypothesis equality holds.NEWLINENEWLINE NEWLINEThe current authors consider the normal order of \(F(n)\) and prove in Theorem 1.1 that for \(\varepsilon> 0\) almost all natural numbers \(n\) satisfy NEWLINE\[NEWLINE(K(n))^{{1\over 2}-\varepsilon}< F(n)< (K(n))^{{3\over 2}+\varepsilon}\text{ with }K(x):=(\log x)^{(\log\log x)(\log\log\log x)}.NEWLINE\]NEWLINE A key ingredient in the proof is a result due to \textit{P. Erdős} and \textit{C. Pomerance} [Rocky Mt. J. Math. 15, 343--352 (1985; Zbl 0617.10037)] concerning the normal number of prime factors of \(\varphi(n)\). Also required (see Lemma 2.1) is an upper bound for the number \(S(x,d)\) of \(n\) such that \(d\mid\varphi(n)\) and \(\varphi(n)\leq x\). Their method also enables the authors to prove that \(\varphi(n)+ 1\) is square-free for almost all \(n\).