Mean divisibility of sequences

Orateur: AKIYAMA Shigeki
Localisation: Université de Tsukuba, Japon
Type: Colloquium de Créteil
Site: UPEC
Salle: P1 P19
Date de début: 18/01/2018 - 15:00
Date de fin: 18/01/2018 - 16:00

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.