Bemerkungen über Fordkugeln. (Remarks on Ford balls) (Q1821144)

For Gaussian integers a,b denote by F(a/b) the Ford ball, i.e. the closed ball in \({\mathbb{C}}\times {\mathbb{R}}\) with center (a/b, 1/2\(| b|^ 2)\) and radius \(1/2| b|^ 2\). \textit{G. J. Rieger} [J. Reine Angew. Math. 303/304, 1-20 (1978; Zbl 0392.10031)] studied the density of the Ford balls near the plane \({\mathbb{C}}\times \{0\}\), more precisely the area m(x) of the intersection of all Ford balls with the unit square in \({\mathbb{C}}\), shifted to height 1/x. For \(x\to \infty\) m(x) tends to 3/(4L(2,\(\chi)\)), \(\chi\) the nonprincipal character mod 4. In the present paper the error term m(x)-3/(4L(2,\(\chi)\)) is sharpened to \[ O(x^{- 1}\quad \exp (-c(\log x)^{3/5}\quad (\log \log x)^{-1/5})) \] resp. assuming a slightly generalized Riemann hypothesis to \(O(x^{- 13/12+\epsilon}).\) Further the author gives an asymptotic formula for the integral mean of the error term. The proofs utilize calculations for the Euler and Möbius function in algebraic number fields. The results can be extended to imaginary quadratic number fields with class number one.