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Grandes valeurs du nombre de factorisations d’un entier en produit ordonné de facteurs premiers
| Content Provider | Semantic Scholar |
|---|---|
| Author | Hernane, Mohand-Ouamar Nicolas, Jean-Louis |
| Copyright Year | 2007 |
| Abstract | 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 $n=q_{1}^{\alpha _{1}}q_{2}^{\alpha _{2}}\ldots q_{k}^{\alpha _{k}}$ , 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 primep−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 C1>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 C2 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 |
| Starting Page | 277 |
| Ending Page | 304 |
| Page Count | 28 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/s11139-007-9028-6 |
| Volume Number | 14 |
| Alternate Webpage(s) | http://math.univ-lyon1.fr/~nicolas/hernanefacto.pdf |
| Alternate Webpage(s) | http://math.univ-lyon1.fr/~nicolas/hernane2.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
