The Riemann Hypothesis, Explained
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- Опубліковано 19 кві 2024
- 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.
00:00 A glimpse into the mystery of the Riemann Hypothesis
01:42 The world of prime numbers
02:30 Carl Friedrich Gauss looks for primes, Prime Counting Function
03:30 Logarithm Function and Gauss's Conjecture
04:39 Leonard Euler and infinite series
06:30 Euler and the Zeta Function
07:30 Bernhard Riemann enters the prime number picture
08:18 Imaginary and complex numbers
09:40 Complex Analysis and the Zeta Function
10:25 Analytic Continuation: two functions at work at once
11:14 Zeta Zeros and the critical strip
12:20 The critical line
12:51 Why the Riemann's Hypothesis has a profound consequence to number theory
13:04 Riemann's Hypothesis shows the distribution of prime numbers can be predicted
14:59 The search for a proof of the Riemann Hypothesis
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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”
lmaoo
One has already been done - and the prize turned down. Fermat's last theorem would have been a Clay Institute award but was solved before the prizes were offered, but Andrew Wiles has received prizes approaching £3 million and a knighthood which isn't so bad really.
did you asked how to the professor?
I like your teacher
Compared to such an achievement, a million dollars feels so trivial it's almost humiliating.
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.
Omg Fermat no! You can’t do that!!
this is the way.
me too
Rh is true because I/2
Pft. Whatever, Fermat.
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.
That's what I call a good trip.
This is stupid - there is no magic man laughing at us - stop with these childish ideas
It' has always been plain that we're dealing with a partialy identified/defined state of existence.
Everything we see are aspects of a whole that we have not yet put together. We know this because reality is currently completely unclear and objectively (essentially) meaningless to us. The fractal isn't yet plotted (It may never be).
When we see the truth of material existence, all answers will suddenly fit together and fill out the description of the whole, seamlessly.
@@ramaraksha01 He never mentioned a magic man.
@@vignesh1065 The universe is inanimate - it is dead - it is not alive to be laughing at us. What he is saying is God created all this and is laughing at us for our stupidity
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)
nice pun
@@emigoldber i dont even think it was intended but it is pretty good
Log function is just a reverse function of exponential function.
(Inverse I mean)
yeah but then other parts of it they just brush over like it's nothing
True, Evelyn. That can be a challenge for truly gifted matematicians- to level down and communicate on ‘lower’ levels. The author shows some pedagogical talent here
Proving the Riemann Hypothesis is probably one of the hardest ways to make a million dollars.
hahaha true i'll be doing forex
investing in gamestop is harder
@@shutup4483 you are right. but would u stop calling it an investment pls XD
@@shutup4483 you are 6 weeks to late
yeah we watched the numberphile video too
Whoever does these animations, massive props to you. These are literally the best math illustrations I've ever seen.
i was just going to add that, until i observed your comment.
well, then you probably don't know 3Blue1Brown
@@hansmeiser32 Both are good at what they are doing
The animations here are really awesome. But 3B1B is still the best.
3blue1brown is worth checking out too!
For the first time in my 46 years, I have truly understood what the Riemann Hypothesis actually is. Thank you!
Fully understood? I'm about your age Mike. When we got to the zero-to-one boundary i went - huh? what? that continued onward.
Never stop learning. Coz people live up to 75 years
@@andyc9902 wait until you hear about 76 year olds
@@whatsoup they should prepare for the death. Unlike 46 year old
I’m 17 and i really want to understand it but i didn’t. Can you please explain what you’ve understood
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!
Be even more amazed when remember that they died before even the electric light was made available to public. Let's not talk then about mechanical calculators...
@@k-force8325 yes, they had what is called "mechanical calculators", which is something like an automated abacus via gears. And they were HUGE (in modern standards) and WEIGHTED a ton... For example, you have the "Pascaline" built by Blaise Pascal, and it was an "Arithmetic Machine" in 1642.
@@franzrogar there were even massive mechanical computers that calculated calculus, much before the small pocket sized scientific calcultors we carry nowdays
@@manavshah8335 I know, I wrote about them in the post I sent 5 months ago before the one you wrote 2 days ago...
I agree, Riemann, Euler, Gauss, and other mathematicians and physicists are truly amazing. Their work has had a profound impact on our understanding of the world, and their discoveries are still being used today.
I'm glad you enjoyed the video! I put a lot of work into the animations, and I'm always happy to hear that people enjoy them.
I think one of the things that makes these mathematicians so special is their curiosity. They were always asking questions and trying to understand the world around them. They were also very creative, and they were able to come up with new and innovative ways to solve problems.
Thank you, Quanta Magazine. My understanding of the Riemann Hypothesis went from 0% to 15%. Great job (I mean it).
15%? sheesh, i guess ur a bit off by about +14.999997%
@@brrrrrrruh Spoken like a true mathematician
@@Artist_of_Imagination true
15%? more like 2% - for me anyway
15% is great.
I'm pretty close to being able to say the name!
hold my beer, I got one A in math in high school, I got this
ur getting the million prize?
no I think the person who will solve this will not drink beer .. but rather some sophisticated tea
Hold my bong water, i got a shocking suprise in math, I've got bees
i got ^ ^
base course -.-
There is already one A in Math.
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
you inspired me, magic man. gonna do the same
@@SublimeWeasel
Hey , How it’s going ?
@@mafhim62 hi. I didn't study math rigorously. Other than that, meh. You?
@@SublimeWeasel
I did , I failed the first three times, but succeeded the fourth!
If you ever need help I am here for you
@@mafhim62thank you. though, what do you mean by "failed the first three times"? what did you even try to do? im now imagining you soving the entire math itself in 4 tries lol
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
It's a bit more complicated than that, though, because exponents have roots as well.
@@jdeep7 idk what that guy is talking about with roots but I guess the complex logarithm isn't a well defined function since there are infinitely many possible imaginary parts for a given input
@@jdeep7 Exponents and powers are often taught in school as the same thing, and the inverse of a power function is a root. Is the reverse of 2^3=8 cbrt(8)=2, or is it log2(8)=3?
right?!
Pretty sure teacher himself did not know that
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
one has to wonder what those people might be able to achieve with modern technology
matplotlib omegalol
@@dwightk.schrute8696 they would probably all use Pascal and create their own framesworks because the other ones, "don't do exactly what I want"
@@wil8785 python?
@@dwightk.schrute8696 Python would make Gauss unstoppable oh my god
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.
100% agree
There's a janitor in Boston who I think could take a crack at it.
Good Will Hunting movie???
Busy eating apples
He’s wicked smaht
@@niceguy3388 Did you like them apples?
Or this patent clerk.
Can I just appreciate how well the animation is? Literally, WOW.
Can I as well?
Not just the animation , the explanation as well
Pieck!
Figuratively, WOW.
No, you cant
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..."
Rieman hypothesis solved by a indian
It solved by telugu man in india
@@wassilywsky6333 omg it WAS solved! That is so amazing
Recently a guy from india solved this
@Chepanu gaka chepanu cambridge university? i thought they were from cambridge, massachusetts?
also according to the clay mathematical institute, the problem is still unsolved and opened. i dont yet have the math skills to evaluate his proof myself, but it seems that his proof is not based solely on analytical mathematics (which is the point of the millenium problems, no?)
Me: It's been a long day, let's watch some light-minded vid.
UA-cam: How bout Riemann Hypothesis?
Same here.
Same
🤣🤣🤣🤣🤣🤣🤣🤣
actualyl same
Oof, true, what am I doing at midnight here, UA-cam?
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!
I was just thinking exactly the same about this video in particular, an and I've watched hundreds of vids and read dozens of books.
it has to do with fourier analysis. because the function with the log of the primes can be written in another way so that the part where you put in the zeta zeros has a cosine. that means that every zero is like a wave. and if you add all those waves together you get this function in 14:26
it is kinda ironic for a musician like me to watch a random math video and hear harmonics mentioned, like what if all the math mental gymnastics is reducible to waves and harmonics ?
Indeed.
@@gardendado1999 I think Pythagorus might have a bone to pic w/ you on that one.
If there was a video like this for every math concept, I would never take my eyes off the computer screen.
Then you'd be dead.
This is how all math concepts should be taught
have you heard of 3blue1brown?
u virgin?
@@miguelcorreia6357
> "u virgin?"
> "Cyka Blyat Man"
A clear and concise presentation on a challenging topic.
Well done! Great animations go a very long way to illuminating the discussion which is as relatively simple and clear as possible. Thank you.
It's not integers it's decimal integers I solved this in high school I was a mathematical genius
This is literally the best video on UA-cam 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.
Yeauh my mind was blown when they shouwed the harmonic sums converging
This video also has some beautiful animations and historical informations. I love to understand math with context and this video makes a great job!
Agree
Not really I felt like this didn't explain much for those with some background in Maths, and is prob still too difficult for those without a background to understand. But can't really blame the video since it's only 15 min long
I wonder what they do with the Riemann hypothesis in quantum physics research...
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.
You are so right!
what's your first language
@@James-un8io Español
I know how you feel. It's very hard to find content as well explained in any other languages (native portuguese speaker)
It makes me sad to think of all the people in the world who don't know English. It's a huge disadvantage that they may not even fully appreciate themselves. There are so many great books and documentaries in English. It's not quite as bad as living in a war torn country with no access to running water or electricity, but still pretty bad in terms of the opportunities that it robs you of.
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
Caralho tu tá em todo lugar
Do Not forget 3blue1brown
It is!
i love Tibees too
"next level" means pop-sci where you learn nothing but feel good about yourself. Downvoted this garbage.
cool man, I think I'll solve this over my lunch break
Did you do it? :P
@@earthling_parth yep, working on it!
My conclusion thus far is that this burger needs more sauce
@@willh69 wow, great progress dude. Let me know when you reach to the state of pineapples and bananas on pizza 😆
Overconfident jokes
@@commentsanitizer7929 OvErCoNFiDeNt JoKeS 😡🤬🥵🥵🥵
Incredible. The reveal when all the harmonics are added in and its the primes is fantastic.
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.
If the math itself isn't enough to bring out your passion, what would attracting a bunch of lay people to the field accomplish? You'd get even more morons like Terry whatshisname who insists on Terryology, where 1*1 = 2, just because he couldn't understand basic grade 1 math.
"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
"The 3 dolar problems that kids play with it?" Hahaha
I think I would want to piss before anything else.
Amazing Tarot Card Reading.
Is Anandi Dhawan Dead/Alive ??
Amazing Tarot Card Reading.
Is Anandi Dhawan Dead/Alive ??
I'd probably want a coffee before tackling anything complicated.
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!
That is the talent only the TRUE professors posess. Feynman and sagan were like this.
Yeah it sounded nice
Bro ur name is math
Being from an engineering background, even I understood the hypothesis. Your video was unbelievably awesome.
Ditto, though for some steps I would have loved rigorous definitions instead of pattern animations .
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! :)
Have you seen Numberphile? That's a pretty fun math's channel.
Oh and Vihart is relly great too.
But this video was indeed really fun, I'm also happy about the discovery :^)
What is the music name at 2:35?
@dota vinkz I don't mean to be rude but you really do not know nothing about maths, maths is all about creativity, there's no blame in being illiterate about maths, but you should really gotta dive deeper than the horrid algorithmic approach that is present in most engineering courses and high school ones. Logic is beautiful, fun and creative and the best examples are Gödel completeness theorem. Maths are beautiful and creative.
@dota vinkz fun is an human emotion encountered when truly performing a task you are best equipped to do so..in one sense fun follows satisfaction ..it has no fixed origin and can be obtained from myriad of sources ..depends on the individual
And maths is creative if you let it be..
@@stower1999 yup I bet in the future if we are successful in creating artificial intelligence constructs ..they would comment human beings being subjective while dealing with objective problems
Probably the clay institute should start adjusting that prize for inflation.
Y'know, if they made it two million dollars, I might just attempt to solve it.
@@jondunmore4268 thanks for the laugh man that got me :D
$1 million is a humiliating amount for answering a problem that defies centuries of effort from the best minds in mathematics and is tied to the foundations of cryptography and quantum mechanics. But that is where the priorities of mankind lie in the 21st century. And if you say otherwise u must be a socialist and against free markets. Yes there are easier ways to make money for sure
@@david50665 the person / team that solves this isn’t going to be motivated by the $1m. Or an increase. Making it $10m or $100m wouldn’t likely make it solved faster.
But you’re right. What normie cares if this is solved? Does it impact their life?
@@codycast I know that but it's a matter of respect and society's priorities...i would prefer if we apply your logic on other fields such as athletes, entrepreneurs, movie stars etc...in theory they should all do it because they love what they do... not because someone throw them a peanut like a monkey...due to market efficiencies, it seems only frivolous work can be well compensated
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.
Edwards has written several great books, not only this one but also books like Galois Theory and Fermat's Last Theorem. They are not easy, but if you put in some work, you'll find the beauty of mathematics. Edwards died November 10, 2020, 84 years old.
This didn't work well for John Nash, he's a crazy quilt. He's weird looking
nothing like Russell Crowe.
Ok fr best comment
Heyy thanks I didn't know this book existed!
Trauma Surgeon
There's another very good book, entitled "Prime Obsession" that alternates chapters on the theory with biographical chapters on Riemann. If you love math, it's a wonderful book. Highly recommended.
I think you deserve $1 million just for explaining this hypothesis in a clear and understandable language. Well done!
Numberphile also did it REALLY well.
3blue1brown has only animated it quite well
Unfortunately, despite the rhetoric, most maths pros, like Riemann himself, really don't want know why R's zeta formula functions as it does, nor why RH remained unsolved for more than 157 years. Also, like Riemann, nor do they want to learn or do anything other than what they are doing inside The Box of the current paradigm of their fave maths niche. If that were untrue Riemann could have solved RH--IFF he could've gotten out of his tumnel-vision syndrome (& outa The Box). Also, if the culture of current maths was not allergic to superior theory & metatheory of maths & logic it would be easy to get my proofs reviewed, published & verified. As is, that's almost impossible. Sigh...seems a shame to let 21 years of good work and next-gen maths go to waste. Oh, well...humanity is clearly stuck with a culture of cowardice, conceit & corruption. So, i guess we're doomed. So, nothing matters. Rite?
@@MichaelMonterey among us
@@jwust1n > Hi. Thanks for noticing. Yet thats a bit cryptic. Care to expand your comment?
By seeing this video the way we think maths varies differently from region to region and place to place that makes it beauty ful
someone give the animators a raise; kept me interested throughout the vid
I like the guy's voice, too. Interesting and not patronizing.
If he narrated my life, I might try.
Keep pumping out content like this. Love the level of detail & creativity in these videos.
Me too. It makes me feel like I'm doing something with my life even though I'm slouching back and passively consuming someone else's hard work.
@@MikhailFederov That’s called passive learning
The best scientific communication video I've ever watched!
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 UA-cam comment section like it was nothing.
true and imagine it gets 0 like and is hidden away forever lmaoo :(
@@aidancanoli welcome to my world.
Will hunting has ENTERED the chat
If some random person solves the problem in the comment section they're most likely full of shit and believe in the levitational properties of mercury.
@@Edeinawc Someone with a real solution would indeed probably prefer another outlet to publish their findings, but I find it amusing to consider the notion of that outlet being this comment section despite better alternatives.
Imagine if Reimann had a computer back then
He would lost himself in cat videos and distracted done nothing.
He would've been on a social networking platform like insta. The man was depressed af man his life was pretty sad. I came to know about him by a book called hyperspace.
Probably make a good fortnite player. Remember Reimann wasn't above average mathematician before college and he wanted to pursue Chemistry.
Could be a big thug life moment for mathematics
Or...
A big bruh moment....
He did have a computer but it was a wetware model.
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.
You got that far, eh?
HA! 39!!!! Whooped your backside!!!! I'm the greatest.......
Weakling, I got 42 seconds in.
I completed the whole video but it's mostly wierd and I have a lot to learn I'm in my 12th grade now
@@klam77 😂😂😂
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.
ANd I still couldn't understand much of anything at all.
@@HitBoxMaster Me neither, although I think I felt the breeze as it went over my head
@@HitBoxMaster I have a math degree and don't understand this hypothesis. The video took a couple of leaps that lost me.
agreed
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.
Same
Could have made it easier by writing as multiples of s. Like 0s 1s 2s etc.
Yeah I had to watch that part a few times to get it, the rest of the video went fine!
@@xiphosura413 The part at 13:14 where he talks about harmonics is where he presents that modified step function and mentions "harmonics" I'm not able to follow anymore. What is he talking about?
Exactly where I got confused.
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?
now that you mention it. i also want one
Unlike many tough math problems, the general consensus is that no one has a clue for how to solve this. Most of the progress that has been made has been to show that it is equivalent to other conjectures, but no one knows how to solve those either. The Wikipedia article has a decent list of some facts, which if proven, would imply the Riemann hypothesis.
It is complicated. There even is this de Branges thing (if somebody who is not a total nutcase writes down a proof attempt and nobody feels like checking it as that would be too much work)
RH feels like a Gödelian Sentence.
@@y__h It can't be. Why? If you prove RH is undecidable, it follows that a counterexample cannot exist, which implies RH is true, which implies it cannot be undecidable.
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
@Keith Smeltz mst-edu haha nice
Counterproof by appeal to authority. Riemann thought it needed a proof, so it needs a proof.
@@xTheUnderscorex :(
Proof by appeal to the stick. If you _don't_ want your sorry butt kicked, then Riemann's hypothesis is true. Q.E.D.
@@whatsthisidonteven Proof by exultation of masochism, I do want my sorry butt kicked so Riemann's hypothesis remains unproven
This guy explaining imaginary numbers made more sense than when I learned about them last year in class
Complex numbers would make much more sense if you were shown it in its most useful applications, such as electric signals, or mechanical movement. In the Euler's formula, you can see as how a complex number can be understood as consisting of two components: one cosine function to depict the horizontal component, and a sine function to depict the vertical component. Imagine a circular movement of a point in that plane. For each point, there is a cosine component, giving you the projection onto the horizontal, and a sine component, giving you the projection onto the vertical.
It boils down to a simple triangular calculation. A point in a plane can be expressed by its Cartesian coordinates, or, by its Polar coordinates.
Consider the imaginary i to map a 90 degree angle on the complex plane. Each time you apply one times i, you move by 90 degrees counter-clockwise. Travel twice 90 degrees (twice i), and you have traveled 180 degrees: you have reached -1. Continue so, and each time you jump one time i, you jump 90 degrees counter-clockwise.
It's what we call "quadrature". It is applied in Fourier analysis and integration, and very practically in decoding movement and speed direction of electric motors. A motor-decoder detects the rotation of the motor's axis by using two detectors that are aligned in such as way as to register the axis movement with a 90 degree difference between both detectors. When the movement signal has stabilised the signal of one detector, the other detector is picking up the change of its signal and triggers the output to switch. The output is always well defined by this design, as both detectors never have an overlapping status of their signals. A quadrature design is very clever.
It is also very useful in the synthesis of complicated signals by mixing a sine and a cosine function, rendering any intended electric signal (as applied in medical ultrasound devices).
Fourier analysis, quadrature applications, they all revolve around that same concept of complex numbers. It's not just mathematical theory, it is very practical indeed.
That's because there are literally a gazilion bad math teachers. This figure was determined using "alkh3myst's conjecture".
@@Guido_XL they make more sense but are still a pain in the butt. It’s so easy to flip a sign.
To be fair, though, it's much easier to understand the general problem as presented in a 16-minute video, where the rigorous proofs are omitted and the details smoothed over, than to understand the technical details or to work with the precision required by a semester-long course.
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?
Someone reply to this comment later to remind me to learn all these terms. It would probably take me about 30 minutes to even comprehend your comment.
Learn all these terms
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.
Trust me when I say you're not alone...
Ooh this is so much fun. You're in for a good time
I still have to use the "the alligator eats the bigger number" for greater and less then symbols.
We are in the right place
Why literally? And by the way, you can not "Google" anything as Google is a proper noun and not a verb.
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.
Sameeee
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.
You're joking right
@@dtp0119 Obviously not.
Let me know how it goes
@@TheodoreServin As I expected it was pretty easy to solve. I won't release the answer though because that would take the fun away from the people still trying to figure this (rather easy) equation out.
Right between breakfast and cold fusion.
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.
Same
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
Not really. A year or two of undergrad (enough to have a vague idea of complex numbers and limits) is sufficient for the content of this video.
Of course there's a lot of content surrounding the Riemann hypothesis (and the zeta function more generally) that this video didn't cover, and much of that absolutely requires a post-doc level grasp of math to fully understand.. but the video only touched on the surface level stuff and that's not nearly as difficult to grasp.
forget about professorship, all you need is to tame the zeta function and play with it, interrogate it or even torture it sometimes, it will spit facts
"No you don't need an advanced degree in mathematics to go on this journey with me." I was lied to!!!
He never strayed beyond high school math. The problem in the US, is that most schools do not teach their students high school math. They are too busy prepping everyone for the exact questions on the standardized tests.
@@princeofcupspoc9073 Thank you for the passive aggressive insult, but I'm from Britain.
@@matbroomfield Probably holds true here as well, mate.
@@Upstreamprovider :-) thanks
@@princeofcupspoc9073 ...what high school did you go to, where they taught complex analysis or wave functions!?!? I'll admit, I'm from the USA, but we only went through infinite series in my high school
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!
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!
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.