On the random variable $\N^r \ni (k_1, k_2, ..., k_r) \mapsto \gcd(n,k_1k_2... k_r) \in \N$

TL;DR

This paper computes the moments and continuous analogue of a random variable involving the gcd of a fixed number and a product of r integers, using elementary methods, extending previous analytical results.

Contribution

It introduces an elementary approach to compute moments and continuous analogues of a gcd-related random variable, generalizing prior analytical findings.

Findings

Computed moments of the gcd-based random variable

Derived its continuous analogue

Extended previous results by Kurokawa-Ochiai

Abstract

We compute the "moments" and its continuous anaougue of the random variable $\N^r \ni (k_1, k_2, ..., k_r) \mapsto \gcd(n,k_1k_2... k_r) \in \N$ by a purely elementary method. This generalizes a result of Kurokawa-Ochiai, which computed its "average" using some analysis involving L-function.