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arXiv · 1711.04257

On multivariable averages of divisor functions

László Tóth·Wenguang Zhai
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Abstract

We deduce asymptotic formulas for the sums $\sum_{n_1,\ldots,n_r\le x} f(n_1\cdots n_r)$ and $\sum_{n_1,\ldots,n_r\le x} f([n_1\cdots n_r])$, where $r\ge 2$ is a fixed integer, $[n_1,\ldots,n_r]$ stands for the least common multiple of the integers $n_1,\ldots,n_r$ and $f$ is one of the divisor functions $τ_{1,k}(n)$ ($k\ge 1$), $τ^{(e)}(n)$ and $τ^*(n)$. Our formulas refine and generalize a result of Lelechenko (2014). A new generalization of the Busche-Ramanujan identity is also pointed out.

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László Tóth, Wenguang Zhai. 2018-11-02. On multivariable averages of divisor functions. https://doi.org/10.1016/j.jnt.2018.04.015

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