Université Paris-Est Université Paris-Est - Marne-la-Vallée Université Paris-Est - Créteil Val-de-Marne Centre National de la Recherche Scientifique

Mean divisibility of sequences

Type: 
Type: 
Site: 
Date: 
18/01/2018 - 15:00 - 16:00
Salle: 
P1 P19
Orateur: 
AKIYAMA Shigeki
Localisation: 
Université de Tsukuba
Localisation: 
Japon
Résumé: 

A sequence $(a_n)$ of integers is called divisible, if $n\mid m$ implies $a_n\mid a_m$. We consider a weaker terminology: "mean divisibility" and give non-trivial examples which satisfy the property. A typical result is
$$
\forall m\geq 1, \forall k\geq 1, \qquad \frac{\prod_{n=1}^{m} {2kn \choose kn}}{\prod_{n=1}^{m} {2n \choose n}}
\in \mathbb{Z}.
$$
We explain the underlying idea of the proof, which involves an interesting statistical behavior of an arithmetic function.