A Calculation of the Number of Lattice Points in the Circle and Sphere

(3) V x) = Hx P (k/2 + 1) P2(x) has been investigated by many celebrated mathematicians and Wilton [11 gives an account of the early work. More recently there have been theoretical investigations of Pk(X) for higher dimensions, particularly by Walfisz [2], whose notation is being followed here. We write P2(x) = O(xc) to mean, in the usual sense, that there exists K such that I P2(x) L/xc 0, and a sequence of values of x tending to infinity, for which I P2(x) I/xc > K that is, the negation of P2(x) = o(xc). Gauss observed that P2(x) = 0(X1/2) Hua [3] has shown that P2(X) = 0(x 1340),