Abstract
In number theory, one often encounters sums of the form Open image in new window Where D is a bounded domain in R k and e(w) =e2πiw. We shall Refer to the case k = 1 as the one-dimensional case, k = 2 as the two- dimensional case, etc. Our objective here is to give an exposition of van der Corput’s method for estimating the sums in (1). The one- dimensional case is well understood. Our knowledge of the two-dimensional case is fragmentary, and dimensions higher than two are terra incognita We shall review the one-dimensional case in Section 2. In Section 3 we will give an outline of what is known and what is conjectured about the two-dimensional case.
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