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visualization - Number of divisors visualized with the QPochhammer function, how to improve performance of code?


I have this code that is originally Jeffrey Stopple's code for the Riemann zeta function in the complex plane. Because I discovered yesterday that the number of divisors can be generated with the $q$-Pochhammer symbol (QPochhammer), and since Mathematica shows a plot of QPochhammer, I thought that plotting it would be fun.


Here is the code that needs improvement:


Show[Graphics[RasterArray[
   Table[Hue[Mod[3 Pi/2 + 
        Arg[Sum[(s + I t)^(n - 1)*(QPochhammer[(s + I t)^(n + 1), (s + I t)]/

             QPochhammer[(s + I t)^(n), (s + I t)]), {n, 1, 100}]], 
       2 Pi]/(2 Pi)], {t, -1.1, 1.1, .05}, {s, -1.1, 1.1, .05}]]], 
 AspectRatio -> Automatic]

And the code for the number of divisors:


CoefficientList[
 Series[Sum[
   x^(n - 1)*(QPochhammer[x^(n + 1), x]/QPochhammer[x^(n), x]), {n, 1,
     104}], {x, 0, 103}], x] (*_ Mats Granvik_,Jan 03 2015*)


1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4,
6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8,
4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2,...

The plot from the first program that I would like to improve:


Divisors from QPochhammer


Already if someone could post a plot with higher resolution, I would be glad. My computer is rather old.



Answer



It looks like you want to plot the phase-only information of a complex function. Using the following helper functions for plotting the phase-only information complex functions:


hue = Compile[{{z, _Complex}}, {Mod[3 π/2 + Arg[z], 

      2 π]/(2 π), 1, If[Abs[z] > 10^-3, 1, 0]}, 
   CompilationTarget -> "C", RuntimeAttributes -> {Listable}];
ComplexPlotC[f_, {x0_, x1_, δx_}, {y0_, y1_, δy_}] := 
  Image[hue[
     f[Outer[Complex, Range[x0, x1, δx], 
       Range[y1, y0, -δy]]]]\[Transpose], ColorSpace -> Hue, 
   Magnification -> 1];
CCompileC[expr_] := 
  Compile[{{z, _Complex}}, Evaluate[expr], CompilationTarget -> "C", 
   RuntimeAttributes -> {Listable}];


You can then compile your function and plot it (I'll only sum 10 terms, rather than 100, as using 100 would take quite a long time):


func = CCompileC[
   If[Abs[z] >= 0.999, 0, 
    Sum[z^(n - 1)*(QPochhammer[z^(n + 1), z]/
        QPochhammer[z^(n), z]), {n, 1, 10}]]];
ComplexPlotC[func, {-1 + 10^-6 RandomReal[], 1, 
  0.003}, {-1 + 10^-6 RandomReal[], 1, 0.003}]

which gives the following:



enter image description here


Here is a 100-term sum picture (open in separate tab to see slightly larger picture):


enter image description here


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