THE RAMANUJAN JOURNAL, Vol. 14, No. 2 (2007), 277-304.

Grande valuers du nombre de factorisations d'un entier en produit ordonné de facteurs premiers

MOHAND-OUAMAR HERNANE
Institut de Mathématiques, Université Houari Boumédienne, BP 32, El Alia, 16111 Bab Ezzouar, Alger, Algeria
Email: hernane_m@caramail.com

JEAN-LOUIS NICOLAS
Institut Camille Jordan, Mathématiques, Université Claude Bernard (Lyon 1), 69622 Villeurbanne cedex, France
Email: jlnicola@in2p3.fr
Homepage: http://math.univ-lyon1.fr/~nicolas/

Received January 6, 2004; Accepted February 14, 2005

Abstract. Among various functions used to count the factorizations of an integer $n$, we consider here the number of ways of writing $n$ as an ordered product of primes, which, if $n = q_1^{\alpha_1} q_2^{\alpha_2} \cdots q_k^{\alpha_k}$, is equal to the multinomial coefficient $h(n) = \frac{(\alpha_1+\alpha_2+\cdots+\alpha_k)!} {\alpha_1! \alpha_2! \cdots \alpha_k!}$. The function $P(s) = \sum_{\mbox{$p$ prime}} p^{-s}$, sometimes called the prime zeta function , plays an important role in the study of the function $h$. We denote by $\lambda = 1.399433\dots$ the real number defined by $P(\lambda ) = 1$. The mean value of the function $h$ satisfies $\frac{1}{x} \sum_{n\le x} h(n) \sim -\frac{1}{\lambda P' (\lambda )} x^{\lambda -1}$. In this paper, we study how large $h(n)$ can be. We prove that there exists a constant $C_1 > 0$ such that, for all $n \ge3$, $\log h(n) \le \lambda \log n - C_1 \frac{(log n)^{1/\lambda}} {\log\log n}$ holds. We also prove that there exists a constant $C_2$ such that, for all $n \ge3$, there exists $m \le n$ satisfying $\log h(m) \ge \lambda \log n - C_2 \frac{(log n)^{1/\lambda}} {\log\log n}$. Let us call $h$-champion an integer $N$ such that $M < N$ implies $h(M) < h(N)$. S. Ramanujan has called highly composite a $\tau$-champion number, where $\tau(n) = \sum_{d\mid n}1$ is the number of divisors of $n$. We give several results about the number of prime factors of an $h$-champion number $N$, about the exponents in the standard factorization into primes of such an $N$ and about the number $Q(X)$ of $h$-champion numbers $N \le X$. At the end of the paper, several open problems are listed.

Keywords: Factorization, highly composite numbers, prime zeta function, optimization

2000 Mathematics Classification: Primary 11A25; 11N37