Research Article

The graphs of exponential sums

J. H. Loxtona1

a1 School of Mathematics, University of New South Wales, Kensington, New South Wales, Australia, 2033.

In [3], D. H. Lehmer has analysed the incomplete Gaussian sum


where N and q are positive integers with N < q and e(x) is an abbreviation for e2πix. The crucial observation is that, for almost all values of N, Gq(N) is in the vicinity of the point ¼(1 + i)q1/2. This leads to sharp estimates of the shape Gq(N) = O(q½).

(Received June 07 1983)

Key Words:

  • 10G10: NUMBER THEORY; Exponential sums; Estimates on exponential sums