The approximation of Chebyshev's function (a logarithmically weighted prime counting function, shown in blue) using the first 250 pairs of nontrivial zeros , of the Riemann zeta function in von Mangoldt's explicit formula:

At each step, the current function (shown in yellow) is modified by adding a waveform whose frequency and amplitude is related to the next pair of nontrivial (complex) zeros in a simple and direct way.

Also available is a similar animation depicting the emergence of the more familiar prime counting function .

Glen Pugh, while at the University of British Columbia, developed a more sophisticated and interactive applet along similar lines. It can currently be viewed here.

[graphics kindly produced by R. Manzoni on request]

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