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

Abstract

We compute the "moments" and its continuous analogue of the random variable $\N \ni l \mapsto \gcd(l,n_1) \gcd(l, n_2) ... \gcd(l, n_k) \in \N$ 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 Soul\'e type zeta functions for Noetherian $F_1$-schemes in the sense of Connes-Consani.