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arXiv · 1604.04508

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

Titus Hilberdink·László Tóth
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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 $f(n)=n^r$, $φ(n)^r$, $σ(n)^r$ ($r>-1$ real), where $φ$ is Euler's totient function and $σ$ 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.

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Titus Hilberdink, László Tóth. 2016-07-26. On the average value of the least common multiple of $k$ positive integers. https://arxiv.org/abs/1604.04508

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