The sound of primes 2
A little animated journey that builds the bridge between the prime counting function and the zeta-function.
For me it was very fascinating and not at all obvious, when I understood the connection between the primes and the zeta-function. I hope that I can share some of this fascination with you.
Together with the first video it should be self-contained and if you know a bit of calculus and complex numbers, it should be possible to reproduce the presented calculations. The video, however, focuses on the visuals. It is really amazing to see, how the psi-function approximation more and more approaches the step function, when more and more zeros are included.
content:
0:00 intro
0:56 result
2:23 overview
2:44 prime counting functions
5:09 zeros
9:25 poles
10:40 the zeta function continuation
12:50 the zeta function product representation
13:30 the proof
16:55 summary
Let me know what you think in the comments.
The links to:
• Analytic Continuation and the Zeta Function @zetamath
contour integrals:
• Complex Analysis: Cauchy's Integral Theorem @qncubed3
The animations are created with blender. It is possible to access blender as a python library. I started to create a manim like framework based on blender as a rendering engine. The obvious advantage is its 3d possibilities.
As a disadvantage obviously, good hardware is required to create the animations in a reasonable amount of time. If you are interested to hear more about it, just leave a message.
#primes #riemannhypothesis #zetafunction #contourintegrals #complexfunctions
#primecountingfunction #zerosofzeta #nontrivialzeros #residue #mangoldtfunction
References:
Riemann's original work:
https://de.wikipedia.org/wiki/Datei:U...
https://www.claymath.org/sites/defaul...
Translation:
https://www.claymath.org/sites/defaul...
A book that helped me a lot to make sense of what was written in Riemann's paper
H. M. Edwards: Riemann's Zeta Function
Mellin Transform:
Handbook of Formulas and Tables for Signal Processing:
http://www.mongroupsydney1.com/3.pdf