On Differences of Two kth Powers of Integers

The arithmetic function rk−(n) counts the number of ways to write a natural number n as the difference of two kth powers (k ≥ 3 fixed). The investigation of the asymptotic behavior of the Dirichlet summatory function of rk−(n) leads in a natural way to a certain error term Pk−(t). In this article we establish a mean-square upper bound and an Ω-estimate for Pk−(t).