Reading John Derbyshire's Prime Obsession and playing around with Python library matplotlib for a course project brought me to plotting Riemann zeta function. The function is defined as follows:

where s is a complex variable, thus s = a + bi. The real part of s, denoted by Re(s), is a; while the imaginary part, denoted by Im(s), is b.

Python library mpmath has implemented this function in zeta and zetazero. Here is an example of the function plot in the complex plane with Re(s) in the range [-10,10] and Im(s) in the range [-5,35]:

 >>> from mpmath import zeta, cplot  
 >>> cplot(zeta,[-10,10],[-5,35])  

• ζ(s) = v
• Re(s) = 1/2
• Im(s) in the range [0.1,50]
• Re(v) = x
• Im(v) = y

I plot the function value on 2D plane in this movie according to the growth of Im(s).

 import matplotlib.pyplot as plt  
 from numpy import arange  
 from mpmath import zeta  
   
 X = []  
 Y = []  
   
 for t in arange(0.1,50,0.1):  
      v = zeta(complex(0.5,t))  
      X.append(v.real)  
      Y.append(v.imag)  
   
 fig = plt.figure()  
 ax = fig.add_subplot(111)  
 ax.plot(X,Y)
 ax.set_xlabel("Re(v)")
 ax.set_ylabel("Im(v)")  
 plt.show()