# Visualization of Exponential Sums

## by Junxian Li

For Math595, Fall 2015

## Exponential Sum

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

References

(Loxton) The Graph of Exponential Sums
(Lehmer) Incomplete Gauss Sums