The true order of magnitude of Lamé rounding error sums

Abstract

Let$\psi (z): = z - [z]\frac{1}{2}$ (z ∈ ℝ). Further let λ denote a large real parameter. We show that for arbitrary real numbersk and α withk>=2.7013 and 0<α≦1,

$$\sum\limits_{0 \leqq n \leqq \alpha \lambda } {\psi \left( {(\lambda ^k - n^k )^{\tfrac{1}{k}} } \right) = \left\{ {\begin{array}{*{20}c} 0 \\ {\Omega \pm } \\ \end{array} } \right\}} \left( {\lambda ^{1 - \tfrac{1}{k}} } \right) (\lambda \to \infty ).$$