How was this made?
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.

**
Click here to view on it's on page** (MP4, 25 MBytes) or
**view it on YouTube!**

If your browser doesn't display the movie, here's what you're missing. A still image for Li

The above shows the polylog on the complex z-plane,
for fixed *s=0.5+i14*, which is not far from
the first non-trivial Riemann zeta zero at *s=0.5+i14.13*.
The image is centered on z=0, and ranges over the interval [-2.5,+2.5]
in the real and imaginary z-axes (the movie uses [-3.5,+3.5]).
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 phase wraps
by a full two pi around the zeros of the polylogarithm.
The zero Li_{s}(0)=0 is clearly visible in the center
of the image. Many additional zeros are arranged on an approximate
circle near |*z*|=1. These accumulate at the branch point at
*z*=1 on the right hand side. The discontinuity from *z*=1
extending to the right is the branch cut. Just underneath the branch
cut a lone zero is visible: this is a Riemann-zero-to-be. As
*s* is slowly changed from *s=0.5+i14* to
*s=0.5+i14.134725*, this zero will slide upwards, and hit the
branch point precisely. This movement is clearly seen in the animation.

Writing *s=0.5+it*, these zeros circle the origin as *t*
changes. Whenever *t* is a Riemann hypothesis zero, these
zeros pass through the branch point at *z=1* on the
complex plane. This can be seen clearly in the movie: observe
what happens as the frame counter passes each RH zero. The first
few are at 14.134725, 21.022040, 25.010858, 30.424876, 32.935062,
37.586178, 40.918719, and so on.

The Riemann hypothesis states that every (non-trivial)
zero will hit the branch point at *z=1* as it loops along
the polylog complex plane. Each zero you see here, on the polylog
complex plane, is a future Riemann zeta zero, in the making. It has
to loop around the origin first, as seen in this movie, and
then hit the branch point at *z=1*. **Every** zero
you see here is a Riemann zero; the RH would be false if any
of them failed to hit *z=1*.

Riemann zeros also pass through the point *z=-1*, but so do
the zeros of the Dirichlet eta: solutions to
*0=2 ^{s}-1*; see, for instance,
frame number

The horizontal discontinuity in the middle, extending to the right,
is a branch cut, extending from the branch point
(essential singularity) at z=+1.
One may perform an analytic continuation through it, to reach other
sheets. A precise form of the continuation, and its monodromy,
is given in the Polylog and Hurwitz
zeta algorithms paper. A more detailed and precise discussion of
the movies shown on this page is given in the
Polylog and Riemann Hypothesis
paper. This paper provides details for the relationship to the RH.
This includes refining the RH into a strong and weak form. This opens
up a new route to proving RH: one would need to show that certain
polylog "varieties" are "entire" in a certain sense. Each "variety"
is given by the solutions of *Li _{s}(z)=0*/ These give
functions

The polylog has two branch points: one at z=+1 and one at z=0, although the z=0 branchpoint is not present on the primary sheet. Both branchpoints are ultimately due to a logarithms appearing in the analytic continuation of the polylog. The animation below shows three sheets. The center is the principal sheet.

The left image shows the sheet that results from winding around z=1 in
a right-handed manner. There are two branch cuts here, one extending
from z=0 and another from z=1 and both have been arranged to extend to
the right. Thus they overlap; this should not cause confusion, as there
is not much going on in this animation. Notice how the first zero
crosses into this sheet, as *s* moves from *s=0.5+i14* to
*s=0.5+i15*. This first zero continues to rotate counterclockwise
on this sheet, and is joined by additional zeros at *s=0.5+i21*,
*s=0.5+i25*, etc. As these rotate, they move onto the next sheet
and the next, disappearing into the branch cut.

The right image shows the sheet that results from winding around z=1 in a left-handed manner. Again, there are two branch cuts here, but this time arranged so that the cut from z=+1 extends to the right, while the cut from z=0 extends to the left. The red-black transitions, indicating a phase change from +pi to -pi, seem to spiral in a bit more tightly. Presumably, these spirals get a bit tighter, with each successive sheet, but otherwise wrap around this branch point, unto infinity. Increasing tau is an unwinding.

Each image is centered on z=0, and runs over the interval [-3.5,+3.5]
in both the real and imaginary directions. Computation requires from 200
to 400 binary digits of precision, depending on the value of *s*.
This is a lower bound on the precision needed to get nice-looking
images; less than this results in severe visual artifacts. Source code
can be found on github: The Anant
math library for analytical number theory, and
Linas' Analytical
Combinatorics/Dynamical Systems Diary.

What is happening inside the unit disk? What is happening off the critical strip? The movie below provides an answer. Most curiously, the zeroes play "Flying Dutchman": as each zero circles around, it appears to get "back in line". (Of course? One might have imagined that it would forever spiral inwards, but this plainly couldn't of happened, as the polylog is very regular at z=0.)

Another curiosity, visible on both the s=0.5+i tau and s=1.1+i tau movies is how a zero leaves behind a shadow, even as it circles onto the next sheet.

Each image is centered on z=0, and runs over the interval [-1.5,+1.5] in both the real and imaginary directions.

What happens if one travels further along the real direction? Here are two movies, from s=0+i14 to s=-10+i14 and another, from s=0+i14 to s=+10+i14. In the first case, the zeros collapse down; in the second case, the zeros flee to infinity.

This allows one to conclude that the zeros of the polylog form a
two-dimensional variety. Labeling each zero with an integer *m*,
there is a function *z _{m}(s)* such that

What happens off the critical line? This is partly explored in an earlier video; but that only looked at the principal sheet. The video looks at the next sheet over, after winding in the right-handed direction. Three panels are shown. In the center, the principal sheet. To the left and right is the continuation to the next sheet. Both show "the same thing", except that the branch cuts are arranged differently. So, the leftmost panel has two cuts: the cut from z=0 extends to the left, and the cut from z=1 extends to the right. The rightmost panel also has two cuts; both are positioned so that they both go to the right (and thus overlap). The upper half-plane for both the left-hand and the right-hand animations are identical; they are both the same sheet, resulting from a counter-clockwise rotation about z=1.

The bottom half-planes on all three animations differ, although, due to "optical" illusions (exponentially small differences), the left and middle bottom halves eventually look the same. The differences can be clearly seen when 0 < tau < 2; after that, the left-bottom and middle-bottom start to look exactly the same (up to exponentially small differences, not visible with naive pixel color-coding.)

The left-most panel plays another trick on the eyes: the cut from z=1 "disappears" visually after about tau=2. It is still there, but the delta across the cut is exponentially small: the delta goes as exp(-tau) and becomes invisible.

Adding to the confusion is the structure of the polylog monodromy: things on different sheets look quite similar to one another. The left-most animation shows this particularly strongly. For example, at 9 < tau < 12, the left-most animation shows a zero emerging from the cut. That zero looks almost identical to what is on the principal sheet. Yet it is not! Where did it come from? Well, from another sheet (glued along the left-hand cut) that is also spewing zeros. If we were to analytically-continue to that sheet, it would look a lot like the principal sheet. But different, as it would have two cuts, not one. The monodromy is quite the mess. There's a lot of duplicitous behavior, here. Caveat Emptor.

Staring at the left-most panel during the begining of the movie (up to about tau=25 or so), it is clear that there is a cut, starting at z=0 and extending to the left. For frames with tau greater than about 30, the portion of the cut between z=0 and z=-1 "fades away", with no visual distinction across sheets. The cut from z=-1, travelling leftwards, remains prominent. This is another optical illusion.

There's also some interesting funny-business going on in the range of tau=0 to 3, where zeros dance around in an unexpected way, before they settle down to their conventional spiralling pattern.

In short, this animation rewards repeated viewing.

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, 2023 Linas Vepstas

Polylogarithm - The Movie!!!
by Linas Vepstas is licensed under a
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