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Zeta Function Plotter

The applet below animates $\zeta(1/2+it)$ for $t \ge 20$ real, where $\zeta(s)$ is the Riemann Zeta Function. See below for some tips on using the applet controls.

Sorry, but the applet doesn't want to run for some reason

For $\Re{(s)}>1$ , $\zeta(s)$ is defined as

$\begin{displaymath} \zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^{s}} . \end{displaymath}$

where the series converges absolutely and uniformly, and is related to the prime numbers via the Euler product formula

$\begin{displaymath} \sum_{k=1}^{\infty} \frac{1}{k^{s}} = \prod_{p~\mbox{\tiny\rm prime }} \left(1-\frac{1}{p^{s}}\right)^{-1}~. \end{displaymath}$

Although $\sum_{k=1}^{\infty} 1/k^{s}$ fails to converge for $\Re(s) \le 1$ , $\zeta(s)$ can be continued analytically to the entire complex plane except for where the zeta function has a simple pole. The famous Riemann Hypothesis is that all zeros of this function lie on the line where $t \in {\mathbb{R}}$ (called the critical line). It is known that there are infinitely many zeros located on the line. Zeros on this line occur where the little square in the animation passes through the coordinate.

About the Applet

The applet uses the Riemann-Siegel formula for computing values of the zeta function. Hopefully the applet controls are more or less obvious; pass the mouse over the various buttons for a short explanation in the ``messages'' area about their function. The current

and $\zeta(1/2+it)$ value are shown in the upper left hand corner of the applet. Here a few tips:

You can adjust the starting value to anything greater than or equal to 20, but once you get up around 1,000,000,000 or so I would be suspicious about the accuracy. The tracing is buggy for up in the tens of millions (it leaves gaps occasionally); not quite sure what the problem is there.
Adjusting will change the difference between sample points, which in effect speeds up or slows down the animation. As increases so does the frequency of zeros, so you may want to set to 0.05 or something smaller.
The trace button ``T'' will produce a nice trace of the path followed by the plot as increases. Click it again to clear the current trace.
The magnifying glasses are for changing the axis scales. The largest scale shows a window from (-20,-20) to (20,20), while the smallest scale shows a viewing window from (-1,-1) to (1,1).

If you spot any booboos or have any comments, please send me some mail.

updated 13:35:46 Fri Jun 20 2008

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