## 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.