Visualization of Exponential Sums

An exponential sum is defined as \[ \sum_{n=1}^N e^{2 \pi i f(n)},\] where f(n) is a real valued function.

A classical example of exponential sums is the complete Quadratic Gauss Sum given by \[ \sum_{n=1}^p e^{\frac{2 \pi i n^2}{p}}=\left\{ \begin{array}{ll} \sqrt{p} & p\equiv 1\bmod 4 \\ i \sqrt{p} & p\equiv 3\bmod 4 \\ \end{array} \right. ,\] where p is a prime number.

D. H. Lehmer has analysed the incomplete Gaussian sum \[ G_q(N)=\sum_{n=1}^N e^{\frac{2 \pi i n^2}{q}},\] where N and q are positive integers with N < q. Here is what they look like in the complex plane.

For different functions, the graphs look very different and can be very random.

Animation