The largest prime factor of Landau's function
Let \(g(n)\) denote the maximal order of an element in the symmetric group \(S_n\) on \(n\) letters. \textit{E. Landau} [Arch. Math. Phys. (3) 5, 92--103 (1903; JFM 34.0233.02)] proved that \(\log g(n)\sim\sqrt{n\log n}\) as \(n\to\infty\). Let \(P^+(g(n))\) denote the largest prime divisor of \(g(n)\). \textit{J.-L. Nicolas} [Acta Arith. 14, 315--332 (1968; Zbl 0179.34804)] proved that \(P^+(g(n))\sim\log g(n)\sim\sqrt{n\log n}\) as \(n\to\infty\). In the paper under review, the authors obtain the following estimates for \[ H(n)= P^+(g(n))/\sqrt{n\log n}. \] For \(n\geq 4\), \(H(n)\leq H(215)= 1.26542463,\dots\) with equality only for \(n= 215\) (Th. 5.1). For \(n\geq 2\), \(H(n)\leq 1+{\log\log n+10.8\over 2\log n}\) (Th. 8.1). For \(n\geq 133\), \(H(n)\geq 1+{\log\log n-2.27\over 2\log n}\); further for \(N\geq 1755\), \(H(n)\geq 1\) (Th. 9.1). The authors also give bounds for \(P^+(g(n))- \sqrt{Li^{-1}(n)}\) (Ths. 7.1 and 9.2). The proof of Theorem 10.1 shows that \[ P^+(g(n))- \log g(n)= \Omega_{\pm}(n^{1/4}(\log n)^{1/4}\log\log\log n). \]
scientific article; zbMATH DE number 6006055
scientific article