On the circle problem with general weight (Q2777524)

Let \(D=\{(x,y)\in {\mathbb R}^2:(x-a)^2+(y-b)^2\leq r^2\}\) with \(a,b,r\in {\mathbb R}\), \(r\geq 1\), and \(\omega \) be a real valued function defined on \(D\). The author proves asymptotic results for \(R(\omega ;a,b;r)=\sum _{(x,y)\in D\cap {\mathbb Z}^2} \omega(x,y)\) with uniform error estimates. The general result is then applied to the case where \(\omega \) is a generalized polynomial [the author's previous papers, Abh. Math. Semin. Univ. Hamb. 68, 117-124 (1998; Zbl 0979.11049) and Math. Slovaca 49, 263-272 (1999; Zbl 0956.11020) dealt with the case when \(\omega \) is a polynomial].