×

On the primitive circle problem. (English) Zbl 0994.11035

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 W. Zhai and X. Cao [Acta Arith. 90, 1-16 (1999; Zbl 0932.11066)]. Bounds for multidimensional exponential sums are the main ingredient of the proof.

MSC:

11P21 Lattice points in specified regions
11L07 Estimates on exponential sums
11N37 Asymptotic results on arithmetic functions

Citations:

Zbl 0932.11066
PDFBibTeX XMLCite
Full Text: DOI