On the primitive circle problem
Let \(V(x)\) be the number of primitive lattice points \((m,n)\) in the circle given by \(m^2+n^2\leq x\), where \((m,n)\) is said to be primitive if \(\text{h.c.f.}(m,n)=1\). Then it is shown, subject to the Riemann hypothesis, that \[ V(x)= \tfrac {6}{\pi}+ O(x^{221/608+ \varepsilon}) \] for any \(\varepsilon> 0\). Unconditionally the best exponent known is only \(1/2\). Previously the best known conditional exponent was \(11/30+ \varepsilon\), due to \textit{W. Zhai} and \textit{X. Cao} [Acta Arith. 90, 1-16 (1999; Zbl 0932.11066)]. Bounds for multidimensional exponential sums are the main ingredient of the proof.
bounds for multidimensional exponential sums
scientific article; zbMATH DE number 1753101