Research: Number Theory

Fractional Derivatives of the Riemann Zeta Function

Legend for complex plot

The video below shows the α-th Grünwald-Letnikov fractional derivative of the Riemann Zeta Function ζ(s) for α between 0 and 10 on -20 ≤ R(s) ≤ 20 and −3 ≤ I(s) ≤ 27.

The hue represents the argument with red representing the positive real direction and cyan the negative real direction, as shown on the right. Brightness represents absolute value, with 0 represented by black and with white representing infinity.

Plots were generated with an implementation of the methods described in Evaluating fractional derivatives of the Riemann zeta function by Ricky Farr, Sebastian Pauli and Filip Saidak in SageMath. The video was complied with ffmpeg.