A counting problem on lattice points in two-, three- and four-dimensional spaces

Abstract

A counting problem on lattice points in n-dimensional space with rectangular coordinate system was considered. Lattice points are all the points having integer coordinates. The distances of these lattice points to the origin O are denoted from small to large as rn,0<rn,1<rn,2<…, where rn,0=0 denotes the origin itself. All the lattice points scatter on a series of concentric spherical surfaces with the center O and the radii rn,k,k=0,1,2,…. The number of lattice points on the spherical surface with the radius rn,k is denoted as Nn,k. Several properties about the sequences rn,k and Nn,k,k=0,1,2,…, were investigated. The relevant generating function was derived in terms of the elliptic theta functions for convenient calculation of rn,k and Nn,k. We proved that for the 2-dimensional case, the number of lattice points is 4 on each circles with the radii satisfying r2,s2=2h,h=0,1,2,…; for the 3-dimensional case, the number of lattice points is 6 on each spherical surfaces with the radii satisfying r3,s2=4h,h=0,1,2,…; for the 4-dimensional case, the number of lattice points is 24 on each hyper-spherical surfaces with the radii satisfying r4,s2=2h,h=1,2,….