This applet has been developed by Raymond Manzoni, based on my suggestions. It is still very much under development ("Version 0.1", as Raymond puts it). Many new features and improvements will be added in due course.

Clicking with your mouse on the blue field, you are able to explore the behaviour of the Riemann zeta function in the critical strip 0 < Re[z] < 1. Using the "Input on critical line only" feature fixes x = Re[z] = 1/2, which (in theory) allows you to manually "find" nontrivial zeros on the critical line.

The vertices of the polygonal line which responds to your mouse correspond to the partial sums in an infinite sum form of the zeta function. Hence the polygonal line "spirals in" on the value of the zeta function at z = x + iy (as selected by mouse-click).

It is very interesting to experiment with the y-scaling to see what happens in higher regions of the critical strip.

Initially, I thought some of the curious images which emerge may have never been seen before, simply because no one had ever thought of looking. I certainly had never seen anything like them. However, by some strange coincidence, on the day after receiving this applet from Raymond, I discovered G. Tenenbaum and M. Mendès France's wonderful little book The Prime Numbers and Their Distribution, which on p.73-75 contains some very similar graphics. The chapter is called "Stochastic Distribution of Prime Numbers" and concerns Cramér's probablilistic model of the primes.

The connection between the images in that book and the ones you are able to generate with this applet may be no more than a superficial resemblance, but I feel it is worth exploring the possibility that there is a mathematical relationship.