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

TL;DR

This paper calculates moments of a gcd-based random variable using elementary methods, generalizing previous results and linking the average to an invariant of finite abelian groups relevant in algebraic geometry.

Contribution

It provides a new elementary approach to compute moments of a gcd-based variable and relates the average to a finite abelian group invariant, extending prior analysis involving L-functions.

Findings

Computed moments and their continuous analogues for the gcd variable.

Established the average as an invariant of finite abelian groups.

Connected the invariant to zeta functions in algebraic geometry.

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.