I've always been fascinated by the fact that the classical [Gauss sum](http://en.wikipedia.org/wiki/Gauss_sum) has absolute value $\sqrt p$, which is exactly what we would expect if we were to interpret the Gauss sum as a random walk. In particular, I have long wondered whether this apparent link between number theory and probability theory is just a curiosity, or whether it is a symptom of a deeper connection. For example, there are many heuristic arguments in number theory that treat primes as being "random." Is there any example of such a heuristic argument that can be made rigorous by using the above observation about Gauss sums?



Can Gauss sums derandomize any heuristic arguments?
The thing is that it is well known that for the quadratic Gauss sums, expressed as an exponential sum rather than with Legendre symbols, the path is very much not a random walk when you plot it in the complex plane. There is plenty of structure visible as approximate Cornu spirals. 


Such sums are not the only Gauss sums, as I know to my cost; and the quadratic case is atypical (perhaps). But from a high-flown point of view, a Gauss sum is a special function in the theory of finite fields (like a Gamma function). There seems to be more mileage in asking about what is special about it.


