Zeta_machine: colour phase plots of zeta functions
I have recently added a function to zeta_machine to generate colour phase plots of the Riemann zeta function, Hurwitz zeta functions and Dirichlet L functions.
For each pixel in the plot corresponding to a complex value, s, the relevant function, z = f(s) is evaluated, and the pixel is coloured according to the values of Arg(z) and |z|, based on the colouring of the unit disk below:
For each pixel corresponding to z = f(s), the value of Arg(z) determines the hue (reading anticlockwise, with red indicating a positive real value for z), and the value of |z| determines the brightness, with black indicating zero and higher values quickly leading to full saturation.
Pixels close to zeros are coloured black. The new colour map function assigns a white colour to pixels corresponding to poles.
I have set out below, three example images, one for each of the key functions featured on this page.
Each image covers the section of the complex plane between -30 ≤ Re(s) ≤ 10 and -30 ≤ Im(s) ≤ 30, with a resolution of 10 pixels per unit. The images have been rotated 90° clockwise such that the imaginary axis runs from left to right.
Riemann zeta function
Hurwitz zeta function, a = 0.75
Dirichlet L function, principal character, modulus = 5
The trivial and non-trivial zeros are clearly visible as small black dots. In the Riemann zeta function image, the non-trivial zeros line up along the critical line. The Hurwitz zeta function image shows that the non-trivial zeros have variable real parts, whereas the Dirichlet L function image shows that the real parts of the non-trivial zeros are either 0 or 0.5.
