In this article we consider the number of lattice points in -dimensional super spheres with even power . We give an asymptotic expansion of the -fold anti-derivative of for sufficiently large . From this we deduce a new estimation for the error term in the asymptotic representation of for .
We prove analogue statements of the spherical maximal theorem of E. M. Stein, for the lattice points Zn. We decompose the discrete spherical measures as an integral of Gaussian kernels st,ε(x) = e2πi|x|2(t + iε). By using Minkowski's integral inequality it is enough to prove Lp-bounds for the corresponding convolution operators. The proof is then based on L2-estimates by analysing the Fourier transforms ^st,ε(ξ), which can be handled by making use of the circle method for exponential sums. As a...