It is a fairly well-known fact that the Gauss sum of a non-trivial character $\chi$ modulo a prime $p$ is always a complex number with an absolute value of $\sqrt{p}$.

In other words, when the Gauss sums of all non-trivial characters are plotted on the Argand plane, they must necessarily lie on the circle with radius $\sqrt{p}$ centered at the origin. Indeed, this is what is seen; however, the distribution of the Gauss sums on the circle is apparently random.

Is there a reason for this apparent randomness? Has the distribution of the Gauss sums on the circle of radius $\sqrt{p}$ been studied previously? Can one say no more about the Gauss sums, other than that they all have an absolute value of $\sqrt{p}$? Is it even reasonable to expect this distribution to have a pattern?

Here are some plots of the Gauss sums on the Argand plane, for $p = 23$ and $p = 7$:

Plot of the Gauss sums for <span class=$p = 23$.">

Plot of Gauss sums for <span class=$p = 7$.">


1 Answer 1


Ernst Kummer made a conjecture like the OP's, but it was disproven in 1978 by J. Patterson and R. Heath-Brown. I found this in the article 'The determination of Gauss sums' by B. Berndt and R. Evans, which has a plethora of information. If the prime gets large enough, then the distribution is uniform. Patterson even made a precise conjecture for the phenomena you have observed, a tendency for the roots to have positive real part. (See formula at end of the paper's section 3.)


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