The Riemann Hypothesis, Explained 2,720,009 viewsJan 4, 2021

1:47 / 16:23 The Riemann Hypothesis, Explained 2,720,009 viewsJan 4, 2021 95K DISLIKE SHARE DOWNLOAD CLIP SAVE Quanta Magazine 485K subscribers The Riemann hypothesis is the most notorious unsolved problem in all of mathematics. Ever since it was first proposed by Bernhard Riemann in 1859, the conjecture has maintained the status of the "Holy Grail" of mathematics. In fact, the person who solves it will win a $1 million prize from the Clay Institute of Mathematics. So, what is the Riemann hypothesis? Why is it so important? What can it tell us about the chaotic universe of prime numbers? And why is its proof so elusive? Alex Kontorovich, professor of mathematics at Rutgers University, breaks it all down in this comprehensive explainer. Read more at Quanta Magazine: https://www.quantamagazine.org/how-i-... 5,159 Comments rongmaw lin Add a comment... zhongning chen zhongning chen 1 year ago My math professor once said, “I’ve know the existence of these math problems for many years. And I assure you, there are a lot easier ways to make a million dollars” 8K SouthBayJay SouthBayJay 1 year ago Literally if my math teacher had just said “logarithms are to exponents what division is to multiplication,” I would have had much less trouble with them. Thanks dude 3.9K Miguel Eduardo Sánchez Miguel Eduardo Sánchez 1 year ago I have watched countless videos about the Riemann Hypothesis, the Riemann's Zeta function, etc. And this is only one that actually explains the connecction between this function and the distribution of prime numbers. The harmonics part has never been explained to me before. Well done, now I can finally truly understand why this is such a big deal for mathematicians. Well done! 755 ClemensAlive ClemensAlive 10 months ago WHO WANTS TO BE A MILLIONAIRE?! Mathematicians: "No thanks..." 1.4K Will H Will H 1 year ago cool man, I think I'll solve this over my lunch break 1.3K Nath Thydanita Nath Thydanita 9 months ago Can I just appreciate how well the animation is? Literally, WOW. 860 Quanta Magazine Chase Denecke Chase Denecke 1 year ago Whoever does these animations, massive props to you. These are literally the best math illustrations I've ever seen. 2.6K Quanta Magazine Rick Sanchez D99 Rick Sanchez D99 10 months ago If this reaches to you, please create a video for each millennium problem with the clarity of this video. A great introductory video to dummies like us. I never knew Riemann hypothesis had a harmonic side. 325 Aquila Aquila 1 year ago This guy explaining imaginary numbers made more sense than when I learned about them last year in class 384 Quanta Magazine Beautiful Korea Beautiful Korea 10 months ago This is the most concise and well-explained Riemann Hypothesis video ever. 59 Quanta Magazine Badly Drawn Turtle Badly Drawn Turtle 1 year ago These videos always talk about everything that rides on the hypothesis being true. I'd like to see a math channel go into detail sometime on what the implications would be if someone disproves the Reimann hypothesis. What sorts of things would need to be reworked? 15 Catman 321 Catman 321 3 months ago (edited) I can imagine one way this problem may be solved but there is an issue What if there was a problem that you could prove works if and only if the reimann hypothesis is true, but then you prove it is true without relying on the reimann hypothesis, instead using a fact already proven, like e is transcendental or something (obviously something more complex than that but this is just an example) The thing is... what would this problem be? It is possible since Euler and Euclid both proved there are infinite prime numbers in different ways, with different logic behind it, but the issue is what would it be here? 4 Artisorak Artisorak 1 year ago (edited) Proving the Riemann Hypothesis is probably one of the hardest ways to make a million dollars. 4.1K Erald .C Erald .C 1 month ago Brilliant explanation. This makes me love math even more. There is so much beauty and mystery in mathematical patterns. 5 Danny Feinberg Danny Feinberg 1 year ago (edited) Thank you for putting together such a high quality, engaging, and fascinating educational video on this topic. I can’t think of any way this video could be any better and, as someone who has limited knowledge in mathematics, I really appreciated the way you explained the concepts so clearly and elegantly!! Bravo!! This is an excellent example for how educational content should be - succinct, elegantly visualized, fascinating!! Thank you!! Beautiful and engaging visualizations of the concepts!! Would you mind me asking what programs were used for the visualizations? 15 Quanta Magazine Nataranja Nataranja 1 month ago It’s so crazy to know the distribution of prime numbers is dependent upon imaginary numbers, things that would ordinarily seem to be completely unrelated. 5 p/e p/e 1 year ago the production quality on this is way too high for it to only have a million views. it explains the subject so well with such a unique art style in such a short amount of time. keep up the good work 39 Paul Pedersen Paul Pedersen 2 months ago Look at 13:21. At each prime p, the value of the function does not increase by log(p), as stated in the video. If that were the case, each step would be larger than the one before. Instead, the steps increase sometimes by large amounts, sometimes by small ones. For example, the step labeled "log(p)" at 13:21 is at the prime 13, i.e., the size of the step is log(13), but the step at 17, which should be log(17), is smaller. It should be bigger. What am I missing? 8 Harshad Salunke Harshad Salunke 1 year ago Reimann, gauss, euler and all other guys did all this stuff without matplotlib😳 I can't even imagine the extent of their hardwork and dedication 1.9K Mike S Mike S 1 year ago Massive props to you for this video. Excellent voice work, animation and music. Re. the content - I learned enough to know that I'd never cut it as a mathematician. But this is about as approachable an explanation as I think anyone could ask for. Thanks for producing this. 12 Quanta Magazine Avasapphic Avasapphic 1 year ago You just made mathematics fun, I understood only half of it but the video was great, glad I discovered your channel! :) 965 Quanta Magazine Ian Grams Ian Grams 1 year ago (edited) I've known of the Riemann Hypothesis for a bit now, but never bothered to try and understand it because I thought it was beyond my comprehension. But wow this video did a great job at explaining what it says, what lead up to it, and what is significant about it. Thank you to all who made this for expanding my understanding! 7 Ezra Steinberg Ezra Steinberg 1 year ago A masterpiece of mathematical explanation! 45 Quanta Magazine Akshay Akshay 1 year ago (edited) This was an amazing video!!! The moment you said " Logarithm function undoes exponentiation , the same way division undoes multiplication" was the moment I actually understood what Logarithms were. It was like a Aha moment. Thank you for making such a beautiful video. It was like listening to a folk lore. 11 Nikhil Desai Nikhil Desai 6 months ago (edited) Thanks for making this video, even though its only touching the surface it was really informative and easy to understand this problem and even harder to solve. 9 REME REME 2 months ago Very well produced and outstanding narration. Well done! 3 Deepstaria Enigmatica Deepstaria Enigmatica 1 year ago Keep pumping out content like this. Love the level of detail & creativity in these videos. 459 Quanta Magazine Marco Cardin Marco Cardin 11 months ago There's still a point to discuss: even if the hypothesis is proven, it doesn't automatically give you a way to calculate all the zeros, so what would really change? 2 Jason Lee KY Jason Lee KY 5 months ago Im so glad I found this video when it was released. Rewatching this now made me appreciate the language of math so much more now that I'm taking a math degree. 4 Horizon Horizon 10 months ago What a beautifully done and informative video. Thank you for making so difficult a subject so clear. I wish you had been my math teacher. 8 Quanta Magazine Bill Bill 1 year ago Thank you for the explanation, which was wonderfully done and visualized. However, the main thing I came away with is a better understanding that my (at best) average math brain is simply not wired to understand (or care) about these higher level abstractions, as I cannot see by the end of the video why this matters at all, outside of the bounty. As proving the hypothesis is essentially impossible because you have to go to infinity to disprove it, and the level of resolution of the problem (already proven out to the billions, apparently) makes it more or less moot. Am I wrong in believing that the functions that rely upon the Reimann Hypothesis are essentially proven in the real world, at the level of our best technologies? Here's an equation: as jazz fusion is to musicians, the Reimann Hypothesis is to mathematicians. That is to say, a technical curiosity that is mainly noise to most people. 1 SAN SAN 1 year ago I don't have words to say, how beautifully explained. From 15:47 to some seconds after it, shows the power of mathematics... 5 Greg Rodd Greg Rodd 1 year ago For those who saw Beautiful Mind, this was the puzzle Nash was working on at the end of the movie. There is a Dover book from Edwards, "Reimann's Zeta Function". 305 pp. The first 25 pages explain Reimann's original 8 page paper. The rest of the book tackles developments since 1859 (up to 1974). Edward's book is presented as a guide to the primary sources. If you saw "The Man Who Knew Infinity", Hardy and Ramanujan also did work related to the conjecture. Turing also worked on the problem, taking a computational approach. Just so you know the competition and how it relates to nerd culture. I get stuck just trying to draw a Greek Zeta. 369 Squeeze Banana Squeeze Banana 1 year ago There was literally about twelve things I didn't understand before the video even got to the actual problem. Felt like when the whole math test builds on the first problem you couldn't solve. 74 Peter Schmitz Peter Schmitz 1 year ago (edited) Wow, the magic of math strikes again. I always feel that tip of the tongue right on the edge of the conscious mind feeling when these things are explicated. Like the mind wants to find a hole and rush in to see the mystery first hand....and perhaps through insight solve the question it poses. 2 Bill Seidel Bill Seidel 9 months ago (edited) I've watched this video and don't distrust math as much. I appreciate how you talk about math as a tool and what makes this interesting, rather than make it a 'I know more than you' lecture 👍 7 Quanta Magazine Arav A Arav A 1 year ago Watching this video just made me wish I could animate and make my own math videos, does anyone have any suggestions for where I can learn to animate like this? The section from 9:05 to 9:37 was just so satisfying to watch! Any suggestions would be very much appreciated :) 3 Chevalier de Balibari Chevalier de Balibari 10 months ago This reminds me of how that "other" notorious problem in number theory was eventually proved by Andrew Wiles. The eventual solution will be the result of the efforts of many, though not "infinitely many" mathematicians. This video explains the problem clearly and succinctly though I had to stop it a couple of times to contemplate the logic and the process. Mathematicians operate on a different plane than the rest of us, perhaps similar to that of Tibetan Buddhism. Both offer understandings of the cosmos. Both are bound by mystery... 1 az az 1 year ago I have discovered a truly marvellous proof of this, but it's much too large for this youtube comment to contain. Therefore it is left as an exercise to the reader. 5.7K Rationalific Rationalific 1 year ago Wow! This was something else! It's amazing how some people are able to come up with these ideas! And this was a great explanation and illustration of them. 2 s3cr3tpassword s3cr3tpassword 1 year ago This is literally the best video on YouTube explaining why the Rieman hypothesis is related to the prime numbers and why proving it is so important. Other videos only briefly mentions that it's important because the 'prime number distribution is encoded in the function', like bruh that doesnt explain it enough. This video also beautifully shows how anaylitcal continuation works. 793 Jack Trainer Jack Trainer 1 year ago If we had had these videos 25+ years ago, the number of math majors in the US would have increased exponentially. This is great content! Quanta Magazine SpizawkDaKizowz SpizawkDaKizowz 1 year ago This is one of the simplest and most cogent explanations of the Riemann Hypothesis I have ever seen. Thank you! joe dasilva joe dasilva 10 months ago (edited) This was by far, the easiest n best explanation I have ever heard about the Riemann hypothesis. Awesome! 1 Robert Schlesinger Robert Schlesinger 1 year ago Excellent video. Very interesting, informative and worthwhile video. A must see for every mathematics and science student. 1 Quanta Magazine Frank McCann Frank McCann 1 month ago Thanks. Wonderful ☺️. When you're working with waves and need figured as we require, a reduction of complexity makes subjects understandable as you've so aptly done. 1 Κωνσταντίνος Καρβουνιάρης Κωνσταντίνος Καρβουνιάρης 1 year ago I know very little about mathematics yet I was able to keep up with this video till the end. That's a rare talent you've got there, explaining such advanced concepts in plain English. Thank you! 489 Srideep Prasad Srideep Prasad 10 months ago This is great content. A crisp concise and beautifully illustrated summary without any mathematical gymnastics.. jack mack jack mack 1 year ago I get it. Superb presentation. I very much appreciate you taking the trouble to explain this problem. I wonder if there can be a proof. And what would it look like? Intuition tells me it would be able to write on a single page. And what field would it be useful? Cryptography, perhaps? Harshavardhan Ponnada Harshavardhan Ponnada 7 months ago "There is only one way to be sure of it and it's the same way the ancient Greeks did their math: rigorous, absolute mathematical proof" - chills! 6 Christian Pauchet Christian Pauchet 1 month ago (edited) Brilliant video, thank you so much for sharing. Maybe the person or machine that manages to solve this problem might have to think in a superior or different dimension. 1 Chengyu Fan Chengyu Fan 10 months ago The animation in the video is so great! What tools or apps do you recommend to make this kind of animation? 5 InterstellarHD InterstellarHD 1 year ago I think you deserve $1 million just for explaining this hypothesis in a clear and understandable language. Well done! 459 Sergiu Anfield Sergiu Anfield 1 year ago finally someone who understands math and can translate and explain in down to earth language. wish my math teacher was half as good at explaining this as the content creator. thank you for the video, i enjoyed it a lot Elon 22l Musk Elon 22l Musk 1 year ago I am a math student myself and I wish I had a teacher like you and I certainly want to teach like this. jose rojas jose rojas 2 months ago Thank you, Quanta Magazine. My understanding of the Riemann Hypothesis went from 0% to 15%. Great job (I mean it). MATHS WISDOM MATHS WISDOM 9 months ago Loved the presentation. You made it easier to understand. Thankyou sir 2 Nic Flatterie Nic Flatterie 9 months ago (edited) These videos are so well done and are so much better then the way I had to learn some of these concepts. It used to be just a mess of fuzzy equations without context or explaining. I guess I was stuck with poor teachers... 1 Amour Learning Amour Learning 1 year ago You don't only achieve immortality in mathematics by solving the Reimann Hypothesis... You also just straight up achieve immortality. 539 Holographic Steve Holographic Steve 1 year ago Great video! and very well explained! Thank you. J S J S 1 year ago If there was a video like this for every math concept, I would never take my eyes off the computer screen. 500 Quanta Magazine Deepa Singhal Deepa Singhal 1 year ago Perfect video. This is how maths needs to be talked about! Thanks! Rai Zen Rai Zen 10 months ago 13:54 This right here ties all the non trivial zeroes of Reiman's Zeta Function and the modified Prime Counting Function. In other words, if even one of the non trivial zeroes were to be found lying outside S=1/2 between S=0 and S=1, then the entire hypothesis is bound to fail...... Thanks for this video. So simple yet so informative. Haven't really put much attention on the Rieman Hypothesis until I saw your video. 1 Andrey Cheremskoy Andrey Cheremskoy 5 months ago It was an exceptional visualization of what is the Riemann Hypothesis. 2 tartarus216 tartarus216 2 months ago Brilliant explanation. This video will help add many different layman’s perspectives on this problem, if only for a few moments, and hopefully help crack it. 1 Adventure Adventure 9 months ago THIS IS HOW YOU TEACH MATHEMATICS!!! MATHEMATICS IS ABSOLUTELY AMAZING, BUT SOME TEACHERS MAKE IT LOOK LIKE IT'S INCREDIBLY DIFFICULT SO MANY FIND IT BORING. HOPE MORE TEACHERS ARE LIKE THIS!!! 15 Set Mason Set Mason 1 year ago hold my beer, I got one A in math in high school, I got this 1.4K nucksfordacup nucksfordacup 1 year ago This felt like my 50 minute lectures in Calc 3 in Uni. I'd understand 1/4th of the way in, and as soon as the prof segued into the topic we were learning that day, I'd be completely lost. Ferociously scribbling the notes; hearing the words she was saying yet not grasping anything. 2 Jeff S Jones Jeff S Jones 1 year ago Exceptionally clear explanation, thank you! OzanBurada OzanBurada 1 year ago Lovely video. IT explains so well. So with Riemann hypotesis, we can just guess where prime is without any big calculations. 1 KpxUrz5745 KpxUrz5745 11 months ago Really superb presentation. Far and away the best I've seen on this topic. 1 Q269 Q269 1 month ago I have to wonder what the differences are when you review a prime number sequence from multiple numerical basis at once. Would the relation between the primes become apparent by observing the adjustments to "change gears" from one base to the next, and then extrapolate that to infinity? 1 Guru da Ciência Guru da Ciência 1 year ago this is next level content 7.3K Quanta Magazine Roland Gerard Roland Gerard 7 months ago This is great content, very well presented. HELLios6 HELLios6 9 months ago Riemann's hypothesis continues to amaze me. He sure found something amazing 1 Enrico Migliore Enrico Migliore 9 months ago Top quality content. Extremely clear. Thanks. Silh S Silh S 2 months ago This presentation is a real treat 🍰🍯🍹 I feel so grateful everytime I watch thisl 🙏 2 Hitesh Pandharkar Hitesh Pandharkar 4 months ago This explained things very intuitively. Thanks for sharing. Pete S Pete S 1 year ago Me: It's been a long day, let's watch some light-minded vid. Youtube: How bout Riemann Hypothesis? 1.1K john john 1 year ago Such a good video!! This is an actual teacher! 2 Tanavat555 Tanavat555 1 year ago I don't usually comment but holy crap, the quality of this video is insane. it's nice to see more easy to understand science/math content popping up. thanks for the hard work. 153 drfpslegend drfpslegend 1 year ago as someone who wants to do research in this area, this was very insightful! Haneul Kim Haneul Kim 6 months ago Golden content and visual explanation. Thank you very much :) Danny Danny 11 months ago Really good video, easy and nice to follow along! ceydaelcin ceydaelcin 1 year ago You make people fall in love with mathematics. I am genuine about this, thank you for the content mate. ProfessorBeautiful ProfessorBeautiful 10 months ago This is beautifully done. There is also a delightful video of the Riemann challenge being explained to an audience including lots of little kids. The speaker makes a connection to musical instruments... and demonstrates on saxes and clarinet! "The Riemann Hypothesis: How to make $1 Million Without Getting Out of Bed" on the YaleUniversity channel. 1 Yerival Polanco Yerival Polanco 1 year ago This is one of the reasons I am so grateful I learned english so young. There are few non english spaces where I can find such great content. 380 Quang Tran Quang Tran 1 year ago Put the 1M$ unsolved problem aside, this is so oddly satisfying to watch! 49 topquark22 topquark22 8 months ago I already knew this material. I enjoyed this video for the pedagogy, the way you explained it. 1 UTUBDZ UTUBDZ 11 months ago Very good explanation and elegant illustration ! Can We know what tools were used to get such incredible content ? 1 Ian Paul Saligumba Ian Paul Saligumba 9 months ago Amazing video 😮😍👏👏👏 Very easy to understand! More videos like this please 1 Stephen Hicks Stephen Hicks 5 months ago Brilliantly explained, thank-you 4grammaton 4grammaton 1 year ago Can we also have a video about why it's so difficult to prove, or rather why it's been so difficult for mathematicians to find the proof thus far? 359 Thomas Gaines Thomas Gaines 1 year ago Absolutely stunning. Felt like 5 minutes I was so enthralled. 2 unionsafetymatch unionsafetymatch 9 months ago That shot of complex functions at ~ 9:40 was amazing 1 Yo Soy Libre Yo Soy Libre 5 months ago Your explanation of the Riemann hypothesis is wonderful! Raj Raj 1 year ago New here, thanks for the video, I gave a thumbs up. I thought your explanation on convergent and divergent series could have been a bit better, but overall the video was good. savetheguineaworm savetheguineaworm 7 months ago This was a beautiful presentation. Thank you. Rizal Purnawan Rizal Purnawan 1 year ago "If I were to awaken after having slept for a thousand of years, my first question would be; 'has the Riemann Hypothesis been proven?'." - David Hilbert 782 michael bailey michael bailey 10 months ago That was so well done, thank you. lol lol 1 year ago This seems pretty easy to solve though, I'll give it a try tomorrow. 138 Quark Sravitrino Quark Sravitrino 7 months ago Top notch quality content.... Thank you for this <3 Alexander Grube Alexander Grube 1 year ago "Riemann was able to rigorously proof that if you add up all the harmonix of the zeta zeros, you get a perfect match to gauss's modified prime counting function" yes of course, i knew that 31 Hans Nelsen Hans Nelsen 5 hours ago (edited) I love the way he says he'll guide you through this in plain language. Incomprehensible to me non math person. Mr.Mustache Man Mr.Mustache Man 1 year ago There's also general dirichlet functions, which are used to prove Dirichlets theorem on arithmetic progressions. 2 Gerardo Piedras Gerardo Piedras 2 weeks ago "The proof of the Riemann hypothesis is trivial and left to the reader as an exercise" - Bernhard Riemann 3 Marcello Capone Marcello Capone 1 year ago There's a janitor in Boston who I think could take a crack at it. 1.8K