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Dirichlet Series Plotter

The applet below animates the partial sums of the Dirichlet series

\begin{displaymath} \sum_{k=1}^{\infty} \frac{1}{k^{s}} \end{displaymath}

 where $s \in {\mathbb{C}}$ is set by the user. The ``N'' value in upper left hand corner of the applet is a count of the number of terms used in the partial sum. The coordinate pair under the N value is the complex number corresponding to the Nth partial sum.
Sorry, but the applet doesn't want to run for some reason
To use the applet, enter the real and imaginary parts of the starting value, click the ``ok'' button, then press the play button. (Pass the mouse pointer over the various buttons to see a description of their function in the ``messages'' area). Press the trace button ``T'' to trace a path in yellow of the successive partial sums. The zoom buttons will increase and decrease the scale.

For $\Re(s) > 1$ the series

\begin{displaymath} \sum_{k=1}^{\infty} \frac{1}{k^{s}} \end{displaymath}

 converges absolutely and uniformly, and defines the Riemann Zeta Function $\zeta(s)$ which 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 $\Re(s) \le 1$, it 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.

Despite the non-convergence of $\sum_{k=1}^{\infty} 1/k^{s}$ in $\Re(s) \le 1$, it is interesting to observe the behaviour of the partial sums $\sum_{k=1}^{N} 1/k^{s}$ as $N$ increases, especially for $0 \le \Re(s) \le 1$ (called the critical strip). In particular, try $s=1/2+it$ where $t$ gives one of the larger zeros of the Riemann Zeta Function, say $t=$860.4107, or $t=$2180.4953 . Give $t=$21880.6004 a try as well for some interesting looking behaviour.

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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