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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 $s=1$ where the zeta function has a simple pole. The famous Riemann Hypothesis is that all zeros of this function lie on the line $s=1/2+it$ 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 $(0,0)$ 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 $t$ and $\zeta(1/2+it)$ value are shown in the upper left hand corner of the applet. Here a few tips:

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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