Counting Lattice Points in The Sphere

Abstract

We consider the error term P 3 ( R ) = # { n ∈ Z 3 : | n | ⩽ R } − 4 3 π R 3 which occurs in the counting of lattice points in a sphere of radius R. By considering second and third power moments, we prove that P 3 ( R ) = Ω ± ( R l o g R ) . An upper bound for the gap between the sign changes of P3(R) is also proved. 1991 Mathematics Subject Classification 11P21.