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

Abstract

We deduce an asymptotic formula with error term for the sum ${\sum }_{n_1,\dots ,n_k\le x}f([n_1,\dots ,n_k])$, where $[n_1,\dots ,n_k]$ stands for the least common multiple of the positive integers $n_1,\dots ,n_k$ ($k\ge 2$) and $f$ belongs to a large class of multiplicative arithmetic functions, including, among others, the functions $f(n)=n^r$, $\phi (n{)}^r$, $\sigma (n{)}^r$ ($r>-1$ real), where $\phi$ 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 $f([n_1,\dots ,n_k])$ is multiplicative.

arXiv comment12 pages