On the random variable $\mathbb{N}\ni l\mapsto \gcd (l,n_1)\gcd (l,n_2)...\gcd (l,n_k)\in \mathbb{N}$

Abstract

We compute the "moments" and its continuous analogue of the random variable $\mathbb{N}\ni l\mapsto \gcd (l,n_1)\gcd (l,n_2)...\gcd (l,n_k)\in \mathbb{N}$ by a purely elementary method. This generalizes a result of Deitmar-Koyama-Kurokawa, which computed its "average" using some analysis involving L-function. We show this average is nothing but the invariant $\mu (A):={\sum }_{a\in A}\frac{1}{|a|}$ for a finite abelian group $A = \prod_{j=1)^k Z/n_j$. In ArXiv-0910.3879v1, this invariant plays an important role in the Soulé type zeta functions for Noetherian $F_1$-schemes in the sense of Connes-Consani.

11 pages; new sections added

References in corpus (1)

• Les variétés sur le corps à un élément