ฝัง
- เผยแพร่เมื่อ 27 พ.ค. 2020
- Today we introduce some of the ideas of analytic number theory, and employ them to help us understand the size of n!. We use that understanding to discover a surprisingly accurate picture of the distribution of the prime numbers, and explore how this fits into the broader context of one of the most important unsolved problems in mathematics, the Riemann Hypothesis.
It is this channel's inaugural video! Welcome everyone, we hope you enjoy what you see here, and let us know how we did in the comments!
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Wow, if "Euler" was alive, thats the kind of video he would produce: Masterpieces for sharing the beautifull of math to the world.
My math degree is from the 70's, The use of computer graphics to visualize mathematics is phenomenal. That the likes if Reimann, Euler and Gauss could do so in their heads even more so. Lovely presentation,
When I clicked on this video, I wasn't sure if I would last the 55 minutes. As it turns out, that was one of the quickest 55 minutes in my experience of TH-cam. Well done!
This was really great! I've seen pieces of this before, but they either go too quickly or I get lost in the careful details of error estimates/complex analysis/whatever that are important if I wanted to work in the field, but obscure the main ideas. Thanks for making this video!
Thank you! For me, it is very important that intuition precede rigor, and in math generally (and number theory specifically) things are rarely presented in that way. My goal in these videos is to provide that intuition and the scaffolding for reading a more rigorous treatment of this content for those interested.
This is the clearest exposition I've seen on this subject on TH-cam. Thanks for your hard work and look flawed to seeing more.
An amazing lecture! I am in my 80s and I have had a lifelong interest in the PNT. This has given me a deeper understanding than anything I have read previously and inspired me to pursue the topic further.
Thanks! Watched it for the second time. Very helpful.
Great stuff! You are the first one who derives where the Li(x) function really comes from. This is hardly explained in books.
You are an awesome individual prof. I did not expect someone to tell in such detail with such quality, about something as beautiful and "complicated" as the Reimann Hypothesis. Thank you professor! God bless u, even though im an atheist
Brilliant. Thank you for taking the time to create such an engaging teaching video that should make the Riemann Hypothesis understandable to even an interested high school student.
This is the best math video I think I have watched, and I have watched hundreds
Superb.... You spend time explaining in the right places. Cheers from England!
32:50 I've always heard that "the density of the primes near X is log(X)", but never the reason why. My mind got blown there. Great video!
I wish there was as much like buttons as there are the zeta zeroes. The video is beyond incredible. I completed studying apostols ANT prolly like 2 or 3 years ago. And I just visited here today, feels like a joy to freshen the memories. Thank you.
You delivered this exposition with flair and judicious dashes of humor, in fairly granular detail. This is a highly recommendable primer on the topic.
This video has so many good sides to talk about that I'm not even gonna attempt it. Please know this work of yours greatly appreciated and is SUPER helpful to amateurs like myself, and I believe to experienced people as well. Thank you!
It's been a long time that i have these kind of feelings while watching maths videos... Thank you
Thank you for this, it's really great! I've never seen this much context about the Riemann Hypothesis before presented in such an understandable way.
Totally awesome! I've been watching RH and GRH videos and this is a gem, it gives some background from a different point of view and helps cement the bigger picture. in addition to "a picture [or graph] is worth a thousand words", I have to say how much I appreciate being told both what we know and what we don't know, what we don't know (or what wasn't covered and left to Grad classes that we weren't told about) was so often overlooked in my Undergrad math classes. Well done!
I rarely comment but when I tell you I've gone years of not understanding what exactly the hypothesis (the complex version) had to do with the Primes - this lecture was a very good step in understanding so thank you very much!
I would really remain grateful to you because of the pleasure, your effort has given me.
awesome video very calm vary clear.. very intuitive ... love it thank u so much .. greetings from Colombia
If we had school teachers like you the RH would be solved by now 🌼
Brilliant video! Please post more!!!!
Came here from Cracking the Cryptic, have been trying to learn number theory during the pandemic so couldn't avoid clicking on a channel with the name 'zetamath' lol. Excellent stuff, have a 500th like!
Really good presentation. ⭐️
Excellent video! I have seen numerous videos on the Riemann hypothesis, but this one definitely came at the subject from a different angle and provided new and important insights. Thanks!
Very interesting and engaging video. Thanks for making it!
great lecture, professor. Thank you. Pl. continue onto the topics from complex analysis
Just found out about this video and this channel. Excellent presentation and one of the best intro to PMT and RH I have seen so far.
Your videos are criminally under viewed.
You do a really good job of explaining things!
This is incredible! Very well done!
Extraordinary video! One of the best on youtube about RH that I've seen so far. Do you know any book or article that has more details about what you shown us? Thank you!
My favorite that I have found is "Riemann's Zeta Function" by Edwards, though be warned it is quite steep quite fast!
One of the best videos Ive seen on the riemann hypothesis. Thanks!!
Really great! Subscribing!
What a spectacular video!
Sir, thank you so much!!
Very well explained. Great stuff!
The explanations are great - thanks for this super video!
Great video! Your channel deserves more than 70 subscribers
Thanks. That was great and really the best lecture for me I ve seen over the topic!
I really liked this presentation. Well Done!
Awesome video professor!
Thanks a lot sir!!
Thanks very much for the explanation, it's very helpful and insightful
As someone who has a math background who watches and reads about the Riemann Hypothesis and PNT you presented it in a different way
Your right, that sometimes mathematical rigor doesn't allow some mathematicians to make simplifications and analogies, even where they are trivial (and tend to 0)
Have to admit, I never heard this explanation of the meaning of the half in the RH, mind blown!
Thanks
Commenting here to bookmark this for later. Thank you for making this video.
Great Video! Very Informative!
great video - please do post a video that connects this video to the "zeros of that complex function"
What an awesome video! Love it! :D
Excellent presentattion. Thank you !
best video ive ever seen
Thank you for this. I've just started to spend some Covid time looking at the Riemann Hypothesis, something I never did at University when I studied Statistic, but I've spent the last few days trying to understand the Zeta Function. For your next video, it would be amazing if you could attempt to explain exactly how the Zeta function, and its zeros at the Real 0.5 critical line (are the actual imaginary Thetas irrelevant?) equates to this convergence at X^(greater than 0.5). I'm now a new sub - thanks again.
I'm headed that direction, explaining that is one of my main goals of the series! Thanks for the subscription!
Totally agree. A lot of video on Complex Analysis mentioned the 1/2 critical line but didn't relate it to the convergence at X^0.5! Many thanks for your inspiration! Look forward to your next series!
Thank you!! You are so good at explaining complicated stuff !!
This is a criminally underrated channel. Please collab with 3b1b
thank you for sharing, learned a lot from this video.
Great video ... Thank you
Thank you so much for this great video
This channel is totally underrated.
Brilliant!!
Thank you so much!!!
Thank you for making this video :)
Excellent.
wait your new videos ! it's amazing how you explain
Great explainer!
nice one !!
Never plot axes without labeling
Use the option of writing on the side of the screen a reminder of what your variables now designate
fantastic job!
finally i got what the Riemann hypothesis is about
awesome video
it's finally out wooohoo
Congratulations on making such an understandable and fun to watch video on this fascinating topic. I really appreciate that you take things at a leisurely pace, and motivate every step of the way. I had about idea about the "simple" (because you motivated and explained it so well) version of Riemann's hypothesis in terms of the order of |π(x)-Li(x)|.
I am definitely looking forward to viewing the rest of the videos in this series!
Just one more thing: as I was watching the video, and you explained that the density δ(x) of the primes around x is about 1/log x, I thought that in this case, the distance between primes at p is about log(p), so you would expect Σ(p≤x)log(p) to be the total distance up to x, i.e. x. This seems a little different from your explanation at the end. Am I wrong?
well done
i did all the youtube things and I know the drill 😢. i luv ur content ❤
love it !
thanku sir.... please make more videos like that..
Great video! Nice explainations, thoughts and also handwriting! Also a rad red pencil ;)
But "guessing" from the graph? Isn't that exact topic home of skewe's number?
All math starts with guessing, but it doesn't end there. I would bet almost every mathematician has a story of a time they were tricked into trying to prove a pattern continued when in fact it didn't. It certainly has happened to me.
This guy requires more subs than he has....
Everything kinda clicked into place for me at 30:45.. All I can say is WOW.
Amazingggg
Very nice video. I got something out of it and I don't even understand it. I watched the whole thing.
Fantastic! I want to know about Zeta function Zeros on the line Real z=1/2
Awesome video. Where did you learn this and what are some online resources(or books) I could look at?
Part of my reason for making this series is that this content is somewhat disparately spread, and most available resources are written to an audience at quite a high level.
As far as online notes go, Keith Conrad has online notes about a lot of these things, and I think they are invaluable!
there are many videos about the RH, It is sad that the Birch and Swinnerton-Dyer Conjecture for example is not explained in any video. or the yang mills.
hello i am in high school i am trying to learn about the zeta function. which fields would you recommend i explore? i have no exp in number theory, though i do know some analysis(RA&CA)
Silverman's A Friendly Introduction to Number Theory would be a great intro number theory text to start with, and would give you the foundation, together with your analysis, to dig deeper into this stuff.
Can you just take derivatives of approximations and still assume they are approximately the same? I don't know, but it sets my spidey senses tingling, so to speak.
I know this result is accurate but that one step kind of set off alarms.
Great video, nonetheless.
This video is intended to be a quick and dirty motivational intro, and certainly you are right to worry. It is somewhat miraculous that here (and in almost all analytic number theory) these kind of operations give you the correct answer.
Hey. Great video. Not sure if i can follow this as I am not that smart. However, i would like to know what the font is called.
Thanks! The font is the standard LaTeX math font, which I believe is called New Computer Modern Roman.
Give us new vídeos!
Factorial is what I use for my wait function on the TI85...
"so how are we going to get a handle on the density of the primes? the method that we're going to use is... the factorial." - what would lead somebody who didn't already know the end result, to take this path?
Point taken that your formula involving li(x) is a better approximation than x/{log x}. But it's no good if you can't evaluate li(x). I find that pi(x)\approx x/{log x - 1 - 1/log x - 3/(log x)^2}.
16:31 But this is v_p(n!), not v_p(n).
I just defined v_p(n) for this video to be the number of p's in n!, since that was all I cared about for this video. Judging by the comments, this was clearly a mistake, since a lot of people have gotten confused by it and thought I did so in error, given its similarity to other notations.
14:13 "If you prove the Riemann Hypothesis is true, that gives you a very specific answer to how big is this error."
Can you tell me what additional knowledge a *proof* of the Riemann Hypothesis will offer that you don't already get from the Riemann Hypothesis itself?
1:13 So you're using "calculus" to include real functions of reals? The way I read others using the word "calculus" it means processes involving differentiation or integration.
We definitely take derivatives and integrals in this (and future) videos quite a bit!
Hopefully whoever is going to win the million dollar prize does so soon while it's still life changing money. With increasing rate of inflation soon the million will just be someone's monthly salary.
Should be v_p(n!) not (n)
In this instance, v_p(n) is just notation choice for the number of times p goes into n! and is used as such throughout the video. I agree looking back on it perhaps I should have chosen something that looks less like the standard notation for p-adic valuation, but I think this would have been hideous if I had written out v_p(n!) everywhere.
32:27 You can't just apply the derivative operator over an approximate equation. That's not a valid transformation in general.