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Psi(x) Plotter

The applet below animates the partial sums (in order of increasing $\vert\rho\vert$) of Von Mangoldt's formula for $\psi(x)$:

\begin{displaymath} \psi(x)=x-\sum_{\rho} \frac{x^{\rho}}{\rho}+\frac{1}{2}\log{\left(\frac{x^{2}}{x^{2}-1}\right)}-\log{2\pi} , \end{displaymath}

where the $\rho$ appearing in the formula are zeros of the Riemann Zeta Function; more on the significance of $\psi(x)$ and $\rho$ below.
Sorry, but the applet doesn't want to run for some reason
To use the applet, enter starting and ending $x$ values and click the ``ok'' button - this sets the viewing range. Don't go below 1.5 or above 5000 for $x$, and don't make the difference between starting and ending values any more than 100, otherwise you can't see the grid or the graph detail. The applet will stop once the first 5000 zeta function zeros are used. Convergence is fairly fast for small $x$ values, and considerably slower once you set starting and ending $x$ values up in the thousands. Place the mouse pointer over the various buttons for a short description of their function in the messages area.

The function $\psi(x)$ above was first introduced by Chebyshev as

\begin{displaymath} \psi(x)=\sum_{p^{n} \le x} \log{p} \end{displaymath}

where the $p$ are the primes (i.e. the function which jumps by $\log{p}$ at each prime power) and is closely related to the distribution of prime numbers. The $\rho$ appearing in the formula are the zeros of the Riemann Zeta Function which lie in the region $0<\Re{(s)}<1$ (called the critical strip), and the sum should be considered as
\begin{displaymath} \sum_{\rho} \frac{x^{\rho}}{\rho} = \lim_{T\rightarrow\infty} \sum_{\vert\Im{(\rho)}\vert<T} \frac{x^{\rho}}{\rho} \end{displaymath}

The prime number theorem states that $\pi(x)$ (the number of primes less than $x$) is asymptotic to $x/\log{x}$, that is,
\begin{displaymath} \lim_{x\rightarrow\infty} \frac{\pi(x)}{x/\log{x}} = 1 . \end{displaymath}

This last statement is equivalent to
\begin{displaymath} \lim_{x\rightarrow\infty} \frac{\psi(x)}{x} = 1 \end{displaymath}

and it was $\psi(x)$ which ultimately led to the proof of the prime number theorem by Hadamard and De la Vallée Poussin (independently) in 1896. The formula for $\psi(x)$ is actually a variation of an explicit formula proposed by Riemann in his ground-breaking 1859 paper On the Number of Primes Less Than a Given Magnitude. Riemann's formula expresses the distribution of the primes in terms of the zeros of the Riemann Zeta Function. It was in this paper that Riemann conjectured the now famous Riemann Hypothesis: all zeros of the zeta function lie on the line $1/2+it$ in the complex plane, $t$ real.

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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