The following code is hacked together from various sources on the web. It is by no means optimal or pretty, but it should get you started.
First you need to pip install: mpmath, sympy, numpy, matplotlib, scipy.
If you run into trouble installing those, you have to install an earlier version of Python and try installing again (I do not recommend virtual environment systems such as conda). I found that Python 3.7.9 installs all of the above, without any problem (on Windows 10).
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
import pylab
import numpy as np
import mpmath
from sympy import (I, oo, Sum, exp, pi, factorial, zeta, im, re)
from sympy.abc import n, x, k
mpmath.dps = 5
#B = oo # infinity
B = 10
bounds = (k,1,B)
Riesz_expr = Sum(
((-1)**(k+1) * (-x)**k)
/ (factorial(k-1)*zeta(2*k)), bounds)
Riesz = lambda z: Riesz_expr.evalf(subs={'x':z})
Max = 2
fig = pylab.figure()
ax = Axes3D(fig)
X = np.arange(-Max, Max, 0.125)
Y = np.arange(-Max, Max, 0.125)
X, Y = np.meshgrid(X, Y)
xn, yn = X.shape
W = X*0
for xk in range(xn):
for yk in range(yn):
try:
z = X[xk,yk] + I*Y[xk,yk]
w = Riesz(z)
w = im(w)
if w != w:
raise ValueError
W[xk,yk] = w
except (ValueError, TypeError, ZeroDivisionError) as exc:
# can handle special values here
raise exc
print (xk, " of ", xn)
# can comment out one of these
ax.plot_surface(X, Y, W, rstride=1, cstride=1, cmap=cm.jet)
ax.plot_wireframe(X, Y, W, rstride=5, cstride=5)
pylab.show()
This took 10-20 minutes to run! Even though I set B = 10 and Max = 2 (the range end points on the 2D domain plane). It produces:
which is the imaginary part of the Riesz function up to some accuracy and within some domain rectangle defined by Max in the code.
What we can conclude is if you want to evaluate the Riesz function and plot it with any sort of speed, you're going to have to do this purely in Numpy or simply switch over to C++ :) I don't know if Numpy has a zeta function or infinite series avaialable, but if not, you have to be creative in how you construct an evaluator for your function using Numpy.
Have fun!
