arcs of the Riemann zeta function’s nontrivial roots

In Boston, we had some rainbows today:

But I want to share with you a different sort of rainbow arc.

I recently got Elias Wegert’s amazing book Visual Complex Functions for my 35th birthday. Along the edges of some of the pages are phase portraits of the critical strip of the Riemann zeta function \zeta(s). Elias has pictures of these strips up on his website. The yellow diagonals are fairly straightforward to see, and he’s written The Riemann Zeta Function on Arithmetic Progressions about them, in which we find out that there is a /stochastic period/ of order 2\pi/log(2).

I used gimp’s zealouscrop function and imagemagick to glue these strips together, yielding this picture:

See those arcs? We don’t know anything about them.

Matt McIrvin’s experiments with SAGE seem to suggest that the arcs are properties of the phase of the zeta function and aren’t immediately apparent if one just has a picture of the roots themselves. The Fourier Transform of Dirac deltas supported at the roots has led to some questions by John Baez:
Quasicrystals and the Riemann Hypothesis
and this post at the n-category cafe: Quasicrystals and the Riemann Hypothesis

As for the arcs, I think one logical stepping stone would be to try to see if other L-functions also have them, and there’s both SAGE’s implementation of the Dokchitser L-functions implementation and the LMFDB to poke around with.


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