Über eine zum Kreisproblem verwandte Summe. II. (On a certain sum related to the circle problem. II) (Q1092098)

Let \(\beta\),\(\gamma\),x\(\in {\mathbb{R}}\) with \(\beta >1\), \(0<\gamma <1/\beta\) and \(x>0\) a large variable. Furthermore, let \(\psi_ k(.)\) denote the 1- periodic Bernoulli polynomial of order \(k\in {\mathbb{N}}\) (k\(\geq 2)\). If \(\beta >2\) then it is known [ibid. 100, 293-298 (1985; Zbl 0568.10024)] that \[ (*)\quad \sum_{n\leq (x/2)^{1/\beta}}\psi_ k((x- n^{\beta})^{\gamma})=O(x^{(1-\gamma)/\beta}). \] The aim of this note is to show that the exponent (1-\(\gamma)\)/\(\beta\) is best possible by determining a principal term in the asymptotic expansion of the l.h.s. of (*). Moreover, this result still remains valid even in the case \(1<\beta \leq 2\), but with a slight restriction for \(\gamma\).