Polylogarithm, The Movie

Come see the Polylogarithm Movie! (19MB) Quicktime! (19MB) YouTube!

A lurid animated movie of the polylogarithm function Lis(z) as s is varied along the critical line s=0.5+it! A raw depiction of the phase of the polylogarithm on the complex z-plane! Watch Riemann zeroes hit thier mark at z=1! See Riemann zeroes leap-frog one-another starting with frame t=48!

See Numerical Algorithms, An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions, Volume 47, Number 3 / March, 2008 pp 211-252.


Here's what you're missing! A still image for Li0l.5+i15(z)

The above shows the polylog on the complex z-plane, for fixed s=0.5+i15, which is not far from the first non-trivial Riemann zero at s=0.5+i14.13. Colors indicate the phase: black indicates a phase of -pi, green a phase of zero, and red a phase of +pi. A sharp red-black transition is just the phase jumping from +pi to -pi. The points around which the phase wraps by a full two pi are the zeros of the polylogarithm. These zeros move about (see the movie!!) as t changes. Whenever t is a Riemann zero, these points pass through the location z=1 on the complex plane (go see the movie!) Riemann zeros also pass through the point z=-1, but so do the zeros of the Dirichlet eta: solutions to 0=2s-1; see, for instance, frame number 2 pi/ln(2)= 9.8.

Details and method of computation are described in detail in the Polylog and Hurwitz zeta algorithms paper. Source code for computing the polylogarithm, the Riemann zeta, and many other functions, can be found in the Algorithmic 'n Analytic Number Theory multi-precision library.
Copyright (c) 2006 Linas Vepstas

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Polylogarithm - The Movie by Linas Vepstas is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.