caa a22 4500 463178092 CHVBK 20180406164831.0 cr unu---uuuuu 170326e20071001xx s 000 0 eng 10.1007/s11139-007-9028-6 doi (NATIONALLICENCE)springer-10.1007/s11139-007-9028-6 Grandes valeurs du nombre de factorisations d'un entier en produit ordonné de facteurs premiers [Elektronische Daten] [Mohand-Ouamar Hernane, Jean-Louis Nicolas] Among various functions used to count the factorizations of an integern, we consider here the number of ways of writing n as an ordered product of primes, which, if $n=q_{1}^{\alpha _{1}}q_{2}^{\alpha _{2}}\ldots q_{k}^{\alpha _{k}}$ , 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)=∑ p prime p −s , sometimes called the prime zeta function, plays an important role in the study of the function h. We denote by λ=1.399433 the real number defined by P(λ)=1. The mean value of the function h satisfies $\frac{1}{x}\sum_{n\leq{x}}h(n)\sim-\frac{1}{\lambda P'(\lambda )}x^{\lambda -1}$ . 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≥3, $\log h(n)\le\lambda\log n-C_{1}\frac{(\log n)^{1/\lambda }}{\log\log n}$ holds. We also prove that there exists a constant C 2 such that, for all n≥3, there exists m≤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 τ-champion number, where τ(n)=∑ d∣n 1 is the number of divisors ofn. We give several results about the number of prime factors of an h-champion numberN, about the exponents in the standard factorization into primes of such an N and about the number Q(X) of h-champion numbers N≤X. At the end of the paper, several open problems are listed. Springer Science+Business Media, LLC, 2007 Factorization nationallicence Highly composite numbers nationallicence Prime zeta function nationallicence Optimization nationallicence Hernane Mohand-Ouamar Institut de Mathématiques, Université Houari Boumédienne, BP 32, El Alia, 16111, Bab Ezzouar, Alger, Algeria aut Nicolas Jean-Louis Institut Camille Jordan, Mathématiques, Université Claude Bernard (Lyon 1), 69622, Villeurbanne cedex, France aut The Ramanujan Journal Springer US; http://www.springer-ny.com 14/2(2007-10-01), 277-304 1382-4090 14:2<277 2007 14 11139 https://doi.org/10.1007/s11139-007-9028-6 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11139-007-9028-6 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Hernane Mohand-Ouamar Institut de Mathématiques, Université Houari Boumédienne, BP 32, El Alia, 16111, Bab Ezzouar, Alger, Algeria aut NATIONALLICENCE 700 1- Nicolas Jean-Louis Institut Camille Jordan, Mathématiques, Université Claude Bernard (Lyon 1), 69622, Villeurbanne cedex, France aut NATIONALLICENCE 773 0- The Ramanujan Journal Springer US; http://www.springer-ny.com 14/2(2007-10-01), 277-304 1382-4090 14:2<277 2007 14 11139 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer