The sound of primes

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.

Let me know what you think in the comments.

Content:
0:00 Intro
0:56 Overview
1:42 Part I: Fourier decomposition of the step function
3:25 Part I: Audio experiment
5:40 Part II: The zeta-function
8:43 Part III: The bridge between the prime counting and the zeta function
12:30 Part IV: Riemann 1859

Corrections:
1:56 should read: - sum_x^p/p
4:54 p relates to x^(1/n), or simpler put p^n = x. The logarithm disappears by the exponentiation

#primes #zetafunction #complexfunctions
#primecountingfunction #zerosofzeta #nontrivialzeros #primedistribution

The material is created with

[sergiokorochinsky]

If there is a million mathematicians in the world, this video should have at least a million views.

[philipm]

I finally understand how the zeta relates to the critical line, what a marvelous explanation! It's so fascinating how a single line can hold so much mystery.

[musescore]

I had stumbled upon this video and saved for later. I must admit the reasoning is fascinating and the graphics and the code in mathematica are very interesting to see. thanks for sharing this video!

[sus]

By far the best explanation of the connection between the Riemann Zeta function and the prime-counting function I have every seen, beautiful work!👍🏽👏🏽

[tim]

Other videos on this site have nice visuals that show how the waves interfere with each other to form the staircase function. Your great contribution was in showing what the equations mean. As you say, let us pause in humble silence as we consider Riemann's gifts, the scope of his knowledge and his soaring imagination.

[ThatHippyPerson]

The best explanation I have ever encountered. It’s like Chinese characters in that a picture can say a thousands words. Such hard work, and you shared. Thankyou

[projectorion]

Stumbled upon this randomly and was very surprised to see a view count of only just over a thousand for such a well-made content! Very interesting video, the animations really do a great job illustrating what is being said.

[KarlFredrik]

Excellent video! Downloaded all 9 pages of Riemann yesterday for fun. Scanned it quickly and realized it's very condensed and would take a fair amount of time to understand. Then I run into your beautiful video! Very good explanation.

[SimonFrank]

Krass heftigst, was so ein Riemann alles im Kopf und von Hand errechnete und vorwegnahm... Ich fasse das nicht wirklich, von seiner Mathematik mal noch ganz abgesehen...!!! Chapeau!

[joelasaucedo]

Wow this video made me so excited to continue studying physics and math. I appreciate the music approach as well. Very beautiful anaolgies and connections you're making.

[LydellAaron]

I really appreciate you linking primes and atoms at the beginning.

[decadentpersian]

The two videos in this series provide the most comprehensive explabations of the Riemann Hypothesis I have ever seen or read. They brilliantly demonstrate the relationship between Zeta Function and Riemann's prime counting function. Magnificent work! As a new subscriber, I look forward to viewing your other work.

[omargaber]

You are a genius in every sense of the word. This channel deserves a million subscribers❤

[Tumbolisu]

This is the first time I have seen the weird infinite product with primes at 12:40. That alone makes the connection to primes so much cleaner. I wonder why other people never mention it?

[lemmaxiom]

Fantastic video, thank you for your work!

[gabrielrhodes]

Wow, absolutely phenomenal video! Thank you so much for all of your hard work

[RSLT]

I love this super-inspiring animation!

[sunritpal]

Incredible video, animation is amazing, your explanation is fantastic 👌

[carlosharmes]

thanks for showing! (specially the link with FourierDataCreation from sound)

[AroundPhysics]

What software you use for producing these wonderful diagrams? And... I appreciate your creativity.