On multivariable averages of divisor functions

Abstract

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

16 pages, revised

References in corpus (2)

• Counting $r$-tuples of positive integers with $k$-wise relatively prime components
• Average number of squares dividing mn

Cited by in corpus (4)

• Hyperbolic summation for functions of the GCD and LCM of several integers
• On certain sums of arithmetic functions involving the gcd and lcm of two positive integers
• Additive arithmetic functions meet the inclusion-exclusion principle: Asymptotic formulas concerning the GCD and LCM of several integers
• Estimates for $k$-dimensional spherical summations of arithmetic functions of the GCD and LCM