Evaluation asymptotique de l’ordre maximum d’un élément du groupe symétrique. (Asymptotic formula for the maximum order of an element of the symmetric group). (English) Zbl 0588.10049

Massias, J. P.; Nicolas, J. L.; Robin, G.

Acta Arith. 50, No. 3, 221-242 (1988).

Let \(S_ n\) be the symmetric group of n elements, and \(g(n)=\max_{\sigma \in S_ n}(order of \sigma).\) E. Landau first considered this function, and proved \(\log g(n)\sim \sqrt{n \log n}.\) Let Li x be a primitive of 1/log x, and \(\omega\) (N) the number of prime factors of N. We prove: \[ \log g(n)=\sqrt{Li^{-1} n}+A(n),\quad \omega (g(n))=Li(\sqrt{Li^{-1} n})+B(n). \] Various upper bounds for the error terms A(n) and B(n) are given, under different possible hypotheses for the error term in the prime number theorem. These upper bounds are shown to be best possible.
The tools are the techniques of the superior highly composite numbers of Ramanujan, explicit formulae in number theory, and the \(\Omega\)-theorems of Landau.

MSC:

11N37	Asymptotic results on arithmetic functions
20C99	Representation theory of groups
11N45	Asymptotic results on counting functions for algebraic and topological structures

Keywords:

symmetric group; number of prime factors; upper bounds; superior highly composite numbers; asymptotic formula; maximum order; arithmetic function