The Distribution of Gauss Sums

Gauss sums are fundamental objects in number theory. Quadratic Gauss sums were studied by Gauss, and after many attempts, Gauss gave a simple formula depending only on the argument of the Gauss sums modulo 4. Higher degree Gauss sums seem to behave differently. Based on numerical evidence, it was suggested by Kummer (1846) that the angles of cubic Gauss at prime arguments are not equidistributed, and exhibit a bias towards positive values. More extensive numerical testing seemed to indicate that the bias does not persist, and that cubic Gauss sums are indeed equidistributed, which was proven by Heath-Brown and Patterson (1979). We will explain in this talk what is involved in proving equidistribution of cubic Gauss sums, in particular why it took more than 130 years after Kummer's observations.

The talk will be accessible to a general mathematical audience, including graduate students.