Über eine zum Kreisproblem verwandte Summe II

Abstract

Let β, γ,x∈ℝ with β>1, 0<γ<1/β and χ>0, a large variable. Furthermore, letψ _k denote the lperiodic Bernoulli polynomial of orderk∈ℕ(k⩾2). If β>2 then it is known [2] that

$$\sum\limits_{n \leqslant (x/2)^{1/\beta } } {\psi _k ((x - n^\beta )^\gamma )} = O(x^{(1 - \gamma )/\beta } )$$

.

The aim of this note is to show that the exponent (1−γ)/β 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<β≦2, but with a slight restriction for γ.