So for New Year's Eve I'm here with some shining stars straight from the Riemann zeta function!
The following plots show the first 5000 non trivial zeros "winding" around with different frequencies. More specifically, what is plotted is the set of points
\begin{equation} (\lambda k\cos(\omega z_k), \lambda k\sin(\omega z_k))_{k=0,\dots,5000} \end{equation} where $z_k$ is the imaginary part of the $k$-th zero and $\omega$ and $\lambda$ are parameters.
Essentially this mechanical setup makes the zeros wind up around the origin while also making them spiral out a bit to see the pattern more clearly.
As these plots show, there appear to be some gaps in the spirals, which should correspond to "holes" in the distribution of the zeros. While tinkering with random points instead of the zeros, the patterns don't show up; while experimenting with equally spaced points (a 1-D lattice) of course the patterns show up whenever the "winding frequencies" are rational numbers.
My question is: are these "empty spirals" just ghosts, oddities generated by some sort of aliasing effect or whatnot, or is there something meaningful to have fun with?
Happy New Year!
EDIT: trying to make the observation more quantitative, I tracked the $x$-coordinate of the barycenter of the cloud of points, and discovered that, plotting this coordinate against the frequency one gets "spikes" in a Fourier fashion, corrisponding to frequencies that are the log of primes or prime powers...here's a plot of that as well (using $\log(\omega)$ instead of just $\omega$ as the frequencies)
