On the average value of the least common multiple of $k$ positive integers

TL;DR

This paper derives an asymptotic formula with error estimates for the sum of a broad class of multiplicative functions evaluated at the least common multiple of k positive integers, extending elementary convolution methods.

Contribution

It introduces an elementary approach to asymptotic analysis of sums involving multiplicative functions of the least common multiple of multiple integers.

Findings

Derived asymptotic formulas with error terms for sums involving LCMs

Extended convolution method to multivariable arithmetic functions

Applicable to functions like n^r, φ(n)^r, σ(n)^r for r > -1

Abstract

We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$ belongs to a large class of multiplicative arithmetic functions, including, among others, the functions $f(n)=n^r$, $\varphi(n)^r$, $\sigma(n)^r$ ($r>-1$ real), where $\varphi$ is Euler's totient function and $\sigma$ is the sum-of-divisors function. The proof is by elementary arguments, using the extension of the convolution method for arithmetic functions of several variables, starting with the observation that given a multiplicative function $f$, the function of $k$ variables $f([n_1,\ldots,n_k])$ is multiplicative.