On differences of two \(k\)th powers of integers
For a fixed positive integer \(k\geq 3\), let \(r^-_k(n)\) denote the number of ways to write a natural number \(n\) as the difference of two \(k\)th powers of (absolute values of) integers. The investigation of the asymptotic behavior of the Dirichlet summatory function of \(r^-_k(n)\) leads in a natural way to a certain error term \(P^-_k(t)\). The author establishes a mean-square upper bound \(\int_ X^ {2X}(P_ k^ -(t))^ 2\,dt\ll X^ 2\) and the lower estimate \(P_ k^ -(t)=\Omega_ +(t^ {1/2}(\log t)^ {1/2k}log\log t)\), where \(X\) and \(t\) are large real variables.
asymptotic behavior of the Dirichlet summatory function
scientific article; zbMATH DE number 1251329