On Certain Exponential Sums Arising In Multiple Fourier Series

Abstract

In the previous paper we found that, nevertheless, one can obtain a variety of results by by-passing a direct estimation of (1.1). It might be worthwhile to observe that a rather complete knowledge of the behavior of the kernels (1.1) would bring with it a solution to some outstanding problems in the theory of numbers. (E.g., take x = 0, 8 = O k = 2, then the sum (1.1) becomes the number of lattice points in a circle of radius R.) In the present paper we shall show that it is possible to obtain sharp estimates for the kernels (1.1) when 8 = (k 1)12. We shall also show how these estimates may be applied to multiple Fourier series; this application will lead to a completion of the solution of the problem of localization first stated by Bochner in [1], and then treated by us in [2]. First let us recall some facts. When k = 1, the kernel (1.1) (when 8 = 0) is the Dirichlet kernel, which because of trigonometric identities has the explicit expression