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