Having browsed this Q on plotting complex functions and the Zeta page, I don't see anything as nice as this plot from Matilde Marcolli's slides "Geometry and physics of numbers".

Looks like brightness and hue reflect Abs and Arg. What are some options to achieve this look and detail?

enter image description here

  • $\begingroup$ Who drew this picture and what software did they use? It appears to be a "standard" Riemann zeta function plot (findable on lots of websites), but to really answer your question you would need to know how much of this is data-driven and how much is gloss and photoshop. $\endgroup$
    – bill s
    Mar 21, 2015 at 19:01
  • $\begingroup$ Brightness is definitely the argument, with black for real numbers and white for imaginary, but the hue doesn't seem to correspond exactly to magnitude, real part, or imaginary part, although they seem close. $\endgroup$ Mar 21, 2015 at 20:54
  • $\begingroup$ Closely related, and possibly a duplicate in the absence of further information: Compiling ColorFunction for faster complex phase-amplitude plots $\endgroup$
    – Jens
    Mar 22, 2015 at 4:41

1 Answer 1


Here's my best guess so far:

fval = ParallelTable[
  Through[{Abs, Arg}[Zeta[x + I y]]], {x, -28, +2, 
   0.05}, {y, -15, +15, 0.05}];

colors = Parallelize@
      arg}, {Mod[abs Sin[arg], 1], (2 Sin[arg]^2 - 1)^11/2 + 1/2, 
      Cos[arg]^2}], fval, {2}];

   colors /. {\[Infinity] | _Interval | Indeterminate -> 0}, 
 ColorSpace -> "HSB"]

enter image description here

I'm not sure why the loops of color go the 'wrong way.'

  • $\begingroup$ Thanks 2012 - this is nice and looks more detailed than the previous efforts, eg as linked by Jens above. I emailed Marcolli re discrepancy but have not heard back. $\endgroup$ Mar 27, 2015 at 2:28

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