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

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.

@Number_Cruncher

2 жыл бұрын

Thank you! Right now, I'm working on part two with hopefully even better animations and more homogeneous presentation. It will focus on Mangoldt's function and will make the relation of the primes to the zeros of zeta more precise.

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

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.

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

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.

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.

@Number_Cruncher

11 ай бұрын

Thank you for your feedback, and in deed, it can be an exciting journey learning about math and physics.

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

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.

@alef0811

Жыл бұрын

Yes back when I first became interested in the Zeta function I thought I’d give the paper a look but wholly cow Riemann really just plows through all the steps so quickly that it was hard to understand how he got from one point to another, not to mention what he did at each step. Coming back to it a few years later I have a general grasp over what he did and I can at least completely understand how he derived the functional equation but I still have a lot of hang ups with the technicalities of the later half of his paper when he relates the prime counting function to the non-trivial zeroes, even if I do understand the techniques that he used. But idk maybe this is just how all mathematicians write papers lol.

@phengyang2639

11 ай бұрын

This is one of the biggest problems with Math in general. Mathematicians always trying to outsmart others so they purposefully leave things out that are critical.

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!

Fantastic video, thank you for your work!

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

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!

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

What a creative video. Great work.

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?

Beautifully explained!

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

@Number_Cruncher

4 ай бұрын

I'm working on it:-)

I love this super-inspiring animation!

Great work!

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.

@Number_Cruncher

Жыл бұрын

But technically speaking, the presentation only marginally touches the Riemann Hypothesis (RH). Almost every statement in the two videos does not depend on the truth of the RH.

@decadentpersian

Жыл бұрын

@@Number_Cruncher True. I think I qualified that in my second sentence. Let me expand on my thinking. As long as one has a basic understanding of complex numbers, it is easy to understand what the problem statement is. I think that is why RH appears so accessible to so many people (compared to the other five unsolved millennium problems. Try explaining P vs. NP to the average person!). What is difficult to understand for mere mortals is the connection between the zeros of the zeta function and the prime numbers. As I said earlier, your videos explain the connections beautifully.

@Number_Cruncher

Жыл бұрын

I totally agree. I remember my own excitement, when finally I managed to understand the connection. Thank you very much for your appreciating feedback.

Nice work, Thank you ^^

Fascinating!

Nice job! Keep it up!

THIS WAS AMAZING!

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

Nice video!

Super Video, vielen Dank!

@Number_Cruncher

2 ай бұрын

Freut mich, wenn es nützlich ist. Danke fürs Feedback

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

@Number_Cruncher

10 ай бұрын

Thank you and I use blender for the visualization. Blender provides a python interface and I created my own animation package that grows while I'm learning more facets of this amazing tool.

Hugs! This is a beautiful video.

Much appreciate the thought and using rainbow colors to show periodicity. If you're taking feedback, I only suggest labelling the chart axis so that the text gets shown as you explain it. This helps the audience look at the same thing you're explaining. That would need extra work on your side, so again: thank you for what you already have done.

Thank you

thank you )

Where does the sound appear?

@Number_Cruncher

11 ай бұрын

It's just an analogy. Because of the underlying Fourier transform, the zeros are kind of tones or frequencies that create the prime counting function when "played" together.

Beautiful animations .. very good explanation.. I request you to make a tutorial video of how you used Blender to make those animations.. I've been trying for long, but have not been able to do so.. It would be very helpful if you make such a tutorial..

@Number_Cruncher

2 жыл бұрын

I tried to explain some of the tools I used for the animations in kzread.info/dash/bejne/gJqHsrqImNy3dNI.html The addon is not overly complicated. It is just a few lines that can easily be customized to one's own needs. The addon is adapted from work done here: m.kzread.info/dash/bejne/Y6Bok5OzppjPdqQ.html Since I didn't find the complex zeta function in the python library, in the video I mainly worked with textures and bump maps calculated with Mathematica. If you only need functions available in python, you can achieve better resolution.

@AmitKumar-xw5gp

2 жыл бұрын

@@Number_Cruncher Thank you for replying and guiding.. It's very helpful.

thanks big teacher

0:53 Hey! He said the thing, “crunching numbers”.

@Number_Cruncher

Жыл бұрын

When I was younger, I only believed in things that I could calculate. Now, I relaxed a bit and believe in things that my cpu can compute.

Correction: Corrections are not for Chapters. (check corrections feature by KZread)

@Number_Cruncher

4 ай бұрын

Thank you for the hint.

Sehr schon.....danke

"turtles all the way down" kind of argument yet for once the crazy was correct

@Number_Cruncher

Жыл бұрын

Yeah, and this without any possibility of numerical verification. All he had, was a slide rule and all he knew, were the first few non-trivial zeros of the zeta-function. What a brave and confident guy.

Ausgezeichnet!

There's that one formula for nth prime, do you think it could shed light onto the distribution?

@Number_Cruncher

Жыл бұрын

Which formula do you mean?

@Gears_AndGrinds

Жыл бұрын

@@Number_Cruncher I think they're talking about Willan's Formula I'm not too sure but I find it pretty interesting

@Number_Cruncher

Жыл бұрын

There is a nice video that demystifies Willan's formula. kzread.info/dash/bejne/nGmnksptYqrMprA.html As far as I remember, it is just a neat mathematical phrase for a simple prime calculation and prime counting algorithm. There is no deeper mathematical concept behind it.

What program are you using at 16:26?

@Number_Cruncher

Жыл бұрын

Mathematica. You can get a trial version for a few days. It's the most powerful computer algebra system that I'm aware of.

@elia0162

Жыл бұрын

@@Number_Cruncher thx,btw one of the best video around on prime distribution

What was the last tool you used to plot that? I'm having trouble reproducing it in python.... Oic manim? Thanks.

@Number_Cruncher

7 ай бұрын

What plot do you mean? Can you write a time stamp?

@maddercat

7 ай бұрын

@@Number_Cruncher The program you were using at the end, 15:55 to 17:00? I think it's manim right? I need to download it.

@Number_Cruncher

7 ай бұрын

In the right panel, it's manim. On the left-hand side, it's Mathematica.

11:22 If the seemless connection between the graph line and video time line don't convert you as a believer of math, nothing will.

@Number_Cruncher

11 ай бұрын

I have to make the video shorter, that the dynamics of the timeline keeps up with the animation 👌

Do you play the sound of the Reimann harmonics at some point? That's what I clicked the video to hear...

@Number_Cruncher

6 ай бұрын

Of course, I do. But it is a visual sound. If played, the Riemann harmonics create the prime counting function. It's a bit abstract, I agree. It's an analogy. The notes are the zeros of the zeta function. And the symphony is the prime counting function.

@iyziejane

6 ай бұрын

@@Number_Cruncher Thanks for the reply! What I mean is that each of the Reimann harmonics will literally have some sound to them when played as an audio waveform. Although they are not periodic, like a a sine wave or square wave etc, you can still interpret the amplitude vs time as a sound (the sound will vary with time since they aren't periodic). Since I see you use Mathematica, you can use the function Sound[ ] to do this, scroll through a few examples in the help and they'll show how to plug in an arbitrary function as the wave form. Most likely they won't sound very nice on the first try, but maybe with the right definition (e.g. scaling the horizontal axis enough to involve higher frequencies frequencies, maybe subtract the overall trend of x/log(x) to make them look more lke functions that oscillate around the horizontal axis, etc ). I might set this up myself tomorrow since I'm curious - I wish the Wikipedia page on decomposing Pi(x) into a sum of li(x^zeta zeroes) has more explicit formulas, so thanks for including more of those in your video.

@Number_Cruncher

6 ай бұрын

Sounds like an interesting experiment, although it is probably difficult to obtain patterns that can be distinguished by ear for different zeros.

@iyziejane

6 ай бұрын

@@Number_Cruncher I've completed my investigation. Since "Reimann harmonics" is not a technical term, I'm actually referring to the ones described in this video by someone else, which look a lot like wave forms (it's similar in content to yours, i.e. the terms that appear for each of the non-trivial zeroes in Reimann's decomposition of the prime counting function) kzread.info/dash/bejne/l2ifscpwoc_ZfZc.html The simplified expressions I used are clearly stated at time 19:23 in that video I just linked (I guess these are the leading asymptotic behavior of the li(x^zero) parts). I put these into Mathematica and listened to the first couple of zeroes, and summing them together. In all cases the sound is like waves on an ocean, where the timing between waves increases over time, and the amplitudes of the waves increases over time as well (i.e. the gaps between primes are getting larger). This stuff I'm talking about likely has no direct mathematical value, but maybe it's a little poetic to listen to this vast ocean of primes. Thanks!

@Number_Cruncher

6 ай бұрын

@@iyziejane I've seen the video and I've seen his expressions. I even asked him in the comments, how he derived them. But he didn't reply. The expressions actually yield the true prime counting function. Would you mind sharing your notebook, I'd really like to dive into the ocean of primes. If you use github, you could make a permanent share and send a link in the comments, so other people could dive into it as well:-)

Why did opening this video try to access my photo library… I have no choice but to report it now

@Number_Cruncher

4 ай бұрын

I have no idea. It has been viewed more than 40000 times without any complaints. Maybe it was a coincidence with another app.

Wher in the viedeo is this sound being played? (The sound of the Zeta function, or anything related, I mean... I cannot find it. At what time in the video exactly do we "hear" the primes?) I just see a lot of well-known facts, and history and even colored pictures, but where is the sound?!

@Number_Cruncher

6 ай бұрын

It is a visual sound. If played, the Riemann harmonics create the prime counting function. It's a bit abstract, I agree. It's an analogy. The notes are the zeros of the zeta function. And the symphony is the prime counting function.

@JosBergervoet

6 ай бұрын

But at what time in the video can I hear it?! Around 4:17 I hear all kind of functions played with the "audacity" program, but nothing related to Zeta, or Pi, or other prime counting related functions... (I guess it must be a "chirping down" sound, for the jumps in the counting become logarithmically sparser in time, correct?)

@Number_Cruncher

6 ай бұрын

@@JosBergervoet In order to hear a sound, you have to have a signal that is periodic in time. If you overlay sine and cosine functions, as shown in audacity, you can for instance approximate a step function. As long as it is periodic, you can hear it with your ears provided the frequency is in a physiological range. The prime counting function however, is not a periodic function. Sent as it is to a speaker, you would not hear anything. This is, where the analogy breaks. There is actually a similar discussion going on with another viewer. She has tried to turn approximations of the "harmonics" provided by the zeros of the zeta function into sound. She also added lines of mathematica code, that turned the functions into sound. But there is nothing in the video, I appologize if the title of the video was misleading.

@JosBergervoet

6 ай бұрын

Now you state incorrect facts! None of the sounds in nature is exactly periodic, and no music is exactly periodic. You, as our favorite number cruncher, can just put any non-periodic function as input in audacity and then we can hear how it sounds. A function without any structure might sound like white noise, with perfect periodicity it would be an everlasting single tone, but most sounds are in between those extremes, @@Number_Cruncher .

@Number_Cruncher

6 ай бұрын

Ah, sorry. I meant oscillating instead of periodic. One needs a signal that oscillates around zero, in order for the membran of the speaker to go back and forth, otherwise there will be no sound waves generated. The prime counting function does not oscillate.

9:39 value of a?

@Number_Cruncher

2 жыл бұрын

There is only the restriction that a>1. This might be surprising at first sight. But it reflects the fact that the integration of complex functions is by in large independent of the precise path.

@simoncopar2512

Жыл бұрын

@@Number_Cruncher How does the rate of convergence depend on the choice of a? Is there some method to the choice of a to make this work faster or better?

@Number_Cruncher

Жыл бұрын

It seems that the convergence improves the closer a is to one.

I would have expected some actual sound!

Yeah you must be straight from the Prussian kingdom of whilhelm Frederick the 199th.

@Number_Cruncher

6 ай бұрын

Can I consider you comment to be an honor? Well, at least you took the time for typing it. I cannot hide my roots, but most likely there are worse things to do than appreciating what Riemann did a long time ago.

There was too little sound.

@Number_Cruncher

11 ай бұрын

Sorry, I'll try harder next time 😉

You forgot to mention that this relation is a theorem which to this date has not been proven! It relies on the assumption that all non trivial zeros lie on a line.

@Number_Cruncher

10 ай бұрын

I don't think so. The calculations performed depend nowhere on the Riemann hypothesis. But I'm not an expert and happy to hear, how your arguments look like.

@falklumo

10 ай бұрын

@@Number_Cruncher I am no expert too. But this is what I learned. The prime step function is only approximated if the Riemann hypothesis holds true. Of course, the calculations are the same, just the result isn't. You wouldn't be able to check numerically though as to this date, the Riemann hypothesis isn't falsified. But maybe, I am wrong and the Riemann hypothesis only allows for tighter estimates.

@Number_Cruncher

10 ай бұрын

Ok, then we just leave it open for discussion. Maybe someone can give a clear answer. I know that you rely on the Riemann Hypothesis, when you want to calculate probabilities. But I don't understand, why the position of the zero should influence the presented calculations. The path of integration is running outside of the critical strip, so the calculation does not rely on the precise real part of the zeros as long as they are located inside the critical strip. I also don't understand, why I shouldn't be able to compare the numerical results with the actual prime distribution function.

@falklumo

10 ай бұрын

@@Number_Cruncher You are probably right. On the numerical part though: if discrepancies exist, then they would emerge at such high values of n that you wouldn't be able to compute. P.S. Thanks for answering. Many KZreadrs actually dont ...

Are you German?

@Number_Cruncher

6 ай бұрын

I cannot hide it.

OMISSION: You have NOT indicated how the Zeta function is analytically continued. How about presenting the ENTIRE Zeta function, especially the portion where the argument is less than +1.

@Number_Cruncher

11 ай бұрын

I'm not sure, I completely understand. (Now, since I understand: I guess you have a point there. But for some reason I found it more useful to not mention this rather technical aspect.)

@DavidFMayerPhD

11 ай бұрын

@@Number_Cruncher Please give me a formula to compute Zera(.25+2i)

@Number_Cruncher

11 ай бұрын

This is actually rather easy. You can calculate it with an alternating series that is derived from the standard definition for real values larger than one. In the wikipedia article on the zeta function en.wikipedia.org/wiki/Riemann_zeta_function#Definition it is explained in the paragraph starting with "The functional equation was established by Riemann in his 1859 paper ...". There you can also find the functional equation that allows to extend the zeta function into the entire complex plane. It is certainly a good point to worry about, since the standard definition breaks down for real values smaller than 1.

@DavidFMayerPhD

11 ай бұрын

@@Number_Cruncher Thank you very much for your illuminating reply.