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”

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.

I really appreciate that you explain the more “basic” things (e.g. what a log function is). It makes the video feel welcoming to people who aren’t necessary very good at math (like me, lol)

This is like becoming an astronaut, discovering a previously unknown planet, finding a river on that planet, and at the bottom of the river is the perfectly fitting other half to a broken rock you found in a river on Earth as a kid. The Universe sees the look on your face and laughs silently.

Proving the Riemann Hypothesis is probably one of the hardest ways to make a million dollars.

Whoever does these animations, massive props to you. These are literally the best math illustrations I've ever seen.

For the first time in my 46 years, I have truly understood what the Riemann Hypothesis actually is. Thank you!

I'm amazed by Riemann, Euler, Gauss and other mathematicians/physicists how their brain and curiousity for math and science managed to find these sort of algorithm and new fundamentals that we even use today. Amazing vid, love your animations!

Thank you, Quanta Magazine. My understanding of the Riemann Hypothesis went from 0% to 15%. Great job (I mean it).

hold my beer, I got one A in math in high school, I got this

When pure mathematics comes with lucid explanations, and the two are complemented by a perfect vanilla icing of aesthetic graphics. A million thanks for this amazing presentation.

Watched this a few months back. A few months of studying maths rigorously later, and I can finally start to appreciate how magnificent this is

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

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

I can't believe I understood this. I've heard about this for years, but this is the first explanation I've seen that makes sense. Great video.

Hats off to Kontorovich sir. He explained such a complicated topic in a very simple manner. I just want to develop this skill.

There's a janitor in Boston who I think could take a crack at it.

Can I just appreciate how well the animation is? Literally, WOW.

This video goes so well with the 3blue1brown one. It explains the Riemann zeta function in more detail and helps you get an actual feel of the 0's, especially the trivial ones. But like all other Riemann zeta function videos I've seen before, they say 'it's important for primes' and refuse to elaborate. NOW I understand, thank you! At least, I understand enough to appreciate it. I've wanted this for so long. Thanks, once again! Also I never appreciated how much of the Riemann hypothesis was actually done by Riemann himself. What a juggernaut! I thought he laid the foundation and it stopped with 'I think the zeroes are on 0.5' and that someone later realised the connection with primes.

I love this video! The explanation of the hypothesis itself is great, with all the right context and friendly explanations, but more importantly I've never seen such a concise yet captivating description of why the hypothesis matters

WHO WANTS TO BE A MILLIONAIRE?! Mathematicians: "No thanks..."

Me: It's been a long day, let's watch some light-minded vid. KZfaq: How bout Riemann Hypothesis?

Just discovering the Quanta math videos. These are my new favorite math explainer videos because - they take on difficult mathematics that I actually want to know about, explain it thoroughly and artfully, with stunning animation that is both entertaining and very well thought out, and makes it all seem easy and inevitable. And having a narrator who has a great voice AND is a personable mathematician seals the deal.

This is a truly fantastic video. The gradual exposition to the very casual person interested in maths is top notch

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!

If there was a video like this for every math concept, I would never take my eyes off the computer screen.

Well done! Great animations go a very long way to illuminating the discussion which is as relatively simple and clear as possible. Thank you.

A clear and concise presentation on a challenging topic.

This is literally the best video on KZfaq 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.

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.

Insanely well animated and absolutely essential to understand the connection between the topics presented. Props!

I've watched many videos on the Riemann Zeta function, but this one is now my favorite. It connects to the primes beautifully. Alex, you've done the world a wonderful service. Thank you!

this is next level content

cool man, I think I'll solve this over my lunch break

Thank you so much! If I had these videos during my mathematics studies it would really be more fun. And surely result in more passion.

Incredible. The reveal when all the harmonics are added in and its the primes is fantastic.

"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

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!

Being from an engineering background, even I understood the hypothesis. Your video was unbelievably awesome.

I loved this video and the math explanations. I could like it 10 times if possible. Great explanation. That is an example of how math should be taught. I am an engineer and at university I had a few good professors, but no professor was as good as this video.

You just made mathematics fun, I understood only half of it but the video was great, glad I discovered your channel! :)

Probably the clay institute should start adjusting that prize for inflation.

The production quality of this content is insane.

As a non-mathematician, I gained so much insight from this one short video! Thank you, thank you, thank you!

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.

I think you deserve $1 million just for explaining this hypothesis in a clear and understandable language. Well done!

The best scientific communication video I've ever watched!

someone give the animators a raise; kept me interested throughout the vid

Keep pumping out content like this. Love the level of detail & creativity in these videos.

By seeing this video the way we think maths varies differently from region to region and place to place that makes it beauty ful

Best explanation of the Riemann Hypothesis explanation I have ever seen! I wish this video existed when I was in college.

Imagine some dude just single-handedly solving this in this KZfaq comment section like it was nothing.

Imagine if Reimann had a computer back then

U guys show video making the same love as you do to math! Thank you! Great video and great explanations

Beautifully made and explained topic, thank you for sharing it with us. What an interesting hypothesis 🤔

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.

I completely followed this for the first 38 seconds.

This is the best video I've watched in a long time. I made my way here from a video about Mertens Conjecture.

This is the most concise and well-explained Riemann Hypothesis video ever.

This is really good. But that moment at 7:13 where he makes the leap to prime numbers went by waaaay too quickly. I had to stop and rewind and pause to catch the transformation.

my knowledge on this factor has went from 0.1% to 5%, good job kind sir.

Great respect to animation as well as speaker..so easily clear the most hardest concept ever..

I studied this hypothesis as a senior math seminar project in undergrad. Very tight and clean synopsis. Wish this video existed back then.

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?

Beautifully lucid presentation. Thank you!

these videos actually make me realise how fascinating maths is

Proof by appeal to authority. If Riemann thought it was true, then it is true. Q.E.D

This guy explaining imaginary numbers made more sense than when I learned about them last year in class

Great animation; even GREATER voice over. Prof. Alex Kontorovich has such a distinctive voice!

After having learned about harmonics and finding the fundamental frequency of functions using imaginary numbers from school, my mind has been blown wide open after seeing this video. I’m having a ‘full circle’ moment right now

Put the 1M$ unsolved problem aside, this is so oddly satisfying to watch!

At some point I didn't understand anything but I kept watching cause the animations are just so crisp

Hi Quanta Magazine Thank you for the wondeful topics you teach us. 🌻 I only understand half of the video could you please explain more about the second half .

Great video and explanation, it really cleared out why its so sought after.

Love the way you illustrate your vids!

If you want to get familiar with the Riemann zeta function, try to proove the following: If you only take every second summand of the zeta function (see 9:42) for a given value of s and draw the intermediate results for the first summands one at a time (similar to 10:03), you get a graph that converges to an outgoing 'spiral' that gets slower and slower, i.e. needs more steps to complete the next rotation than the previous one. You can draw one spiral for the summands with odd 'index' and one for those with even 'index'. Try to proof: 1. The centers of the two spirals will be at different points unless the input (s) is one of the zeros of the original function. 2. The centers never exactly meet for any other input. 3. The centers only meet at the origin (0, 0). 4. There is no input for which only one of the centers is the origin. You might need to find a useful definition/formula for the center first. You might need to exclude trivial cases for some of these. Visualizing this first by plotting the graphs and playing around with the parameter s might be useful. Try to plot both spirals in the same plot. Try flipping the signs to align them. You can assume the Riemann hypothesis to be true if you need it for a proof. Some easier tasks for warm up: a) Which formula describes the length of the nth summand of the zeta function? In other words: What is the distance in the complex plane between two consecutive intermediate results, i.e. between the results for the first n-1 summands and the first n summands? b) Which formula describes the angle of the line segment between those two points in the complex plane? c) Can you use this to formulate the zeta function with two dimensional vectors and without complex numbers? d) What if you only take the summands with odd/even index?

This is one of the most beautiful things I've ever seen! And I am someone who has been horrible at mathematics and failed a bunch of times! But Now I'm working from three years in Computer Science and starting to learn maths from scratch. It is surprising to me how awesome mathematics is if only I could find the right tools or people to learn from.

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

I literally had to google "what is a prime number?" I'm clearly in the wrong place.

Excellent, informative video with truly knockout graphics.

great video. i can tell a lot of work was put into it. I saw someone on tik tok taking the first minute of your video and using it for views without giving you proper credit

I might only understand 10% of this, but I'm still utterly fascinated.

This is the best video explanation of the Riemann Hypothesis. Thank you for taking the time and effort to produce it.

What a beautiful presentation. Thank you. I've subscribed.

Fascinating and informative. If only we could pass the current state of knowledge back to the masters. What would they think or did they just know. 😮. Thank you

This seems pretty easy to solve though, I'll give it a try tomorrow.

A masterpiece of mathematical explanation!

Thank you Alex: before this I didn’t really understand the Riemann Hypothesis. Fascinating video, many thanks.

it was fascinating listening to you professor ! you got a new sub

Can't put it in words how beautifully my brain circuited near the end.

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.

This is so mind blowing I'm scared of truly understanding it in fear of losing my mind

You just made a real Master Class of Math. Thanks! I enjoyed it as much as good music!

"Don't need to be a maths professor to follow" - yeah but no, it helps

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!

Very nice video. Despite being fairly mathematical, I happened to never technically encounter the Reimann hypothesis or taken the time to really understand it until I watched this video.

Thank you for this excellent explanation of how much I don't understand the Riemann hypothesis!

"No you don't need an advanced degree in mathematics to go on this journey with me." I was lied to!!!

What a beautiful explanation of such an advanced topic. I’m an engineer so I have some understanding of higher level math concept, but I showed my girlfriend (a graphic designer) and she understood it too. Thank you

This channel should reach 1 million.👍🏼 What a content,nicely explained.