#dirichlet l function

[primepatterns]

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.

[primepatterns]

After my last post, a Redditor with the user name Dgafaboutconvergence asked me to try to generate some iteration fractal images based on Dirichlet L functions.

These functions can be evaluated as linear combinations of Hurwitz zeta functions, which meant that I could reuse some of my existing code.

After consulting the relevant source code of mpmath I adapted the code of hurwitz_zeta_machine to evaluate and iterate Dirichlet L functions with different modulus values.

I tested the resultant application, inevitably called dirichlet_zeta_machine, and the above image summarises the initial results. Each image covers the section of the complex plane between -6 < Re(s) < 6 and -25 < Im(s) < 25 with a resolution of 10 and 10 significant figures. From top to bottom, the relevant Dirichlet L function moduli are 1 (Riemann zeta function), 3, 5, 7 and 11. The images on the left show the zeros and the images on the right show the images overlaid on the fractal images.