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 address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences . In particular, we obtain explicit expressions for the number of solutions, where ’s are squares modulo . In addition, we obtain expressions for the number of solutions with order restrictions or with strict order restrictions in some special cases. In these results, the expressions for the number of solutions involve Ramanujan...
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...