Über k-freie Zahlen als Summe von zwei Quadraten. (On k-free integers as sums of two squares) (Q1119688)

The author generalizes a result of \textit{K.-H. Fischer} [Arch. Math. 33, 150-154 (1979; Zbl 0407.10038)] on the average order of representations of square-free numbers as sums of two squares to k-free numbers. Let A(x;k) denote the number of solutions of \(1\leq u^ 2+v^ 2\leq x\), \((u^ 2+v^ 2)\) k-free. Let \(\alpha\) be the exponent in the error term of the circle problem. It is known that \(1/4\leq \alpha <1/3\). Then it is proved that \[ A(x;k)=A_ kx+O(x^{1/k} \log x)\quad for\quad k=2,3, \] \[ A(x;k)=A_ kx+O(x^{\alpha})\quad for\quad k>3. \]