Critical Strip Explorer (version 0.67)
This applet has been developed by Raymond
Manzoni, based on my suggestions. It is still very much under
development. New features and improvements
will be added in due course, and there may still be bugs to be removed.
Clicking and dragging with your mouse on the blue field, you are able to
explore the behaviour of the Riemann zeta function in the complex plane.
The origin always appears in the centre of the screen, but by using the
upper-horizontal sliding controller, you can zoom in or out to view a
larger or smaller section of the plane.
The lower-horizontal and right-vertical sliding controllers can be
used to change the x- and y-offsets. These determine
a translation relating the points you click on to the actual values
input as arguments of zeta. You can thus use them to explore
zeta(s) for s far removed from the visible part of the
complex plane. The actual values displayed of course tell you which value
of s you select by clicking on the origin. Also, by sliding these controls, you are able to
effectively slide your point s horizontally or vertically from
its last input value.
If you reset to the default value (s =
1/2 + i14.1347251..., the 'first' nontrivial zero), and then
use the right-vertical controller, you can watch what happens at various
points on the critical line Re[s]=1/2. The upper-horizontal
controller can be used to zoom out and get a wider perspective. The
motion of the polygonal line is not very smooth, you may find, but this
can be adjusted with the left-vertical contoller (output scaling).
Moving this up produces a more gradual, smooth motion.
Constraining the value of x to be
Re[s] = 1/2 will allow you to explore the behaviour of zeta on
the critical line, and (in theory) to manually "find" nontrivial zeros on the critical line.
As well as the x and y constraints, there are also the
possibilities of constraining the phase and the modulus of z,
in order to explore the behaviour of zeta along rays and around circles
centred at the origin.
The vertices of the polygonal line which you see
correspond to the partial sums (or products) in an infinite sum (or
product) expression for the zeta function. There are a number of
options in the lower-left combobox (not currently visible in the IE browser):
- Finite zeta sum is the most familiar expression for zeta(s),
converging for Re[s] > 1. Obviously we can only approximate the
infinite sum with a finite number of terms. This is the case for all options, but you
are free to choose the number of terms ("number of points").
- Mirror zeta sum combines the functional equation
with the usual sum to produce an expression which converges for Re[s] < 0.
- Alternate zeta sum makes use of a well-known technique for
extending the usual sum into the critical strip (i.e. values of
zeta(s) where Re[s] > 0) involving a sum whose terms have
- Mirror alternate sum combines the functional equation with the
"alternate zeta sum" in order to produce an expression which converges
when Re[s] < 1.
- Finite Euler product is based on the usual product-over-primes
form of zeta, also convergent when Re[s] > 1.
- Documentation on the Riemann-Siegel function and Riemann-zeta
function options will appear shortly.
- Riemann zeta 3-D aims to introduce 'perspective' to produce
a 3-dimensional effect. This is still under development.
Hence, with each of the first five options, the polygonal line "spirals in" on the value of
the zeta function at the selected s = x + iy.
It is very interesting to experiment with the vertical scaling
and see what happens in higher regions of the critical strip.
Raymond's recommendations for newcomers:
- select the kind of sum, product or function you want to see in the
- use the up and down arrows of the scrollbar at the right (not the
other parts which scroll "too fast") to scroll the picture
Glen Pugh has produced a couple of related applets:
a "Dirichlet Series
Animation" depicting "wandering partial sums" and
one which plots
the behaviour of zeta along the critical line. Jan van Delden has produced another applet based on Pugh's.