# Sunset at the Sea of Primes

The pictures show a colorful visualization of a complex function, whose nulls are located at all primes on the real axis. The formula is shown below.

Links to related videos:

http://youtu.be/pkJ0GoFsBqs

http://youtu.be/trOkyzhb2RY

http://youtu.be/MDx5W4MaIcI

## Formula

- f(z) = \left| 2 - \sum\limits^\infty_{k=1} \frac{1}{k} \frac{e^{2 \pi i z}-1}{ e^{2 \pi i z / k}-1} \right|

## Sunset at the Sea of Primes

## Sunset at the Sea of Primes (detail)

## Ray of Primes 2

Primes at the right hand side can be identified by a blue “flame” overtopping the other integers.

## Prime Spiral 1

Inspired by Ulam’s prime spiral: the ray of primes warped to an Archimedes spiral.

## Prime Spiral 3

## Prime Spiral 4

Here, the same distance between numbers and between spiral arms was chosen. This leads to a chaotic distribution of primes.

## Prime Spiral 5

Here, the distance between numbers are pi times of the distance between spiral arms. Now primes are forming interesting patterns.

## Prime Spiral 7

Here, the distance between numbers are pi/3 times of the distance between spiral arms. Again, primes are forming interesting patterns.

## Prime Spiral 8

Here, the distance between numbers are pi/8 times of the distance between spiral arms. Again, primes are forming interesting patterns.

## Prime Spiral 10

Here, the distance between numbers are 3.19 times of the distance between spiral arms.