Average number of squares dividing mn

TL;DR

This paper investigates the asymptotic behavior of a sum involving a divisor function over products of integers, deriving an asymptotic formula with a specific error term using advanced complex analysis techniques.

Contribution

It provides a new asymptotic formula for the sum of a specialized divisor function over products, with an explicit error term, advancing understanding in multiplicative number theory.

Findings

Derived an asymptotic formula with an error term of O(x^{10/7})

Applied multidimensional Perron formula and complex integration methods

Enhanced understanding of divisor sums over products of integers

Abstract

We study the asymptotic behaviour of $\sum_{m,n\le x} \tau_{1,2}(mn)$, where $\tau_{1,2}(n) = \sum_{a b^2 = n} 1$, using multidimensional Perron formula and complex integration method. An asymptotic formula with an error term $O(x^{10/7})$ is obtained.