Lattice Points in Many-Dimensional Balls

Abstract

Let Pk(n) be the difference between the number of points of the integer lattice contained in the ball \( {y}_1^2+\cdots {y}_k^2\le n \) and the volume of this ball. The paper investigates the asymptotic behavior of

the sums

\( \sum_{n\le x}{P}_k(n)\kern0.5em \left(k\ge 4\right),\kern0.5em \sum_{n\le x}{P}_3^2(n),\kern0.5em \sum_{n\le x}{P}_4^2(n)\kern1em as\kern0.5em x\to +\infty . \)