On composite integers \(n\) for which \(\varphi(n)\mid n-1\)
The authors give an improved upper bound for the number \(\#{\mathcal L}(x)\) of composite integers \(n\leq x\) that satisfy the condition \(\varphi(n)\mid n-1\). The problem was inspired by a question of \textit{D. H. Lehmer} [Bull. Am. Math. Soc. 38, 745--751 (1932; Zbl 0005.34302; JFM 58.0158.01)] who asked if any such numbers exist. While the answer is not known to date, the fact that no such number was found suggests that \(\#{\mathcal L}(x) \ll x^{o(1)}\) may hold. Present authors show that \[ \#{\mathcal L}(x) \ll x^{1/2}(\log x)^{-1/2+o(1)}. \] This is an improvement over previous results of this type, including [\textit{W. D. Banks} and \textit{F. Luca}, Acta Math. Sin., Engl. Ser. 23, No. 10, 1915--1918 (2007; Zbl 1215.11094)] who showed that \(\#{\mathcal L}(x) \ll x^{1/2}(\log\log x)^{1/2}\), and [\textit{W. D. Banks}, \textit{A. M. Güloğlu} and \textit{C. W. Nevans}, Integers 8, No. 1, A59, 8 p. (2008; Zbl 1232.11095)] who obtained \(\#{\mathcal L}(x) \ll x^{1/2}(\log x)^{-\Theta+o(1)}\) for some \(\Theta\simeq 0.1294\). The main tool developed and used in the present paper is an upper bound for the number of integers \(n\) for which the contribution of a given set of primes to the factorization of \(\varphi(n)\) is relatively small.
scientific article; zbMATH DE number 6108233
scientific article