On the average order of the gcd-sum function over the set of square integers

The gcd-sum function is one of the most important functions that has been studied by many researchers in recent years (Broughan, Bordellès, etc.). The gcd-sum function appears in a specific lattice point problem, where it can be used to estimate the number of integer coordinate points under the square root curve. In this paper, we give an average order of the gcd-sum function over the set of squares.