The Key to the Riemann Hypothesis - Numberphile
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- 게시일 2024. 04. 27.
- L-Functions are likely to play a key role in proving the Riemann Hypothesis, says Professor Jon Keating from the University of Bristol.
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Mathematicians: can't solve Riemann hypothesis
Also mathematicians: *best we can do is to come up with infinitely many more unsolved Riemann hypotheses*
😂😂
Lol yeah
To explain why mathematicians do this: If you can engineer new Riemann hypotheses, you can study them as a whole. It's sort of like if we found life from beyond Earth, we would be able to study what "life" even is and how it can be. Now, if we can somehow solve one of these engineered ones, maybe we can use that solution to somehow solve the original one. At least we would have a framework for how it could be done.
I do appreciate the joke though! I just thought it's valuable to understand exactly why this stuff is important. Mathematicians don't like to assume something is special, that too has to be proven :)
7:16 - Only in math can you get infinitely many rare things.
+boenrobot Yes. Things that are only "countably infinite" (like the integers) are rare compares to the reals (which are "uncountably infinite").
+boenrobot It helps us come a little closer to appreciating the vastness of infinity. You can have numbers spread so far apart and yet still have an infinite number of them.
Example: Set each number to be the size of a power tower. So
n1 = 10
n2 = 10^10 = 10 billion
n3 = 10^10^10 = a number with 10 billion digits
...
and we can fit in an infinite number of these n values...
assuming an infinite universe there are infinitely many sets of infinitely many rare things
+boenrobot Not really, if our universe is infinite, then life would still be rare and could aswell, be present on infinitly many planets.
If throw a coin infinitly many times, I have infinitly many occasions of 1000 heads in a row, still they are rare.
+Baron ultra paw Observable universe is finite and contains a finite number of particles (somewhere on the order of 10^80 particles and 9 photons per particle). About 10 billion galaxies with 100 billion stars per galaxy. Infinity is really only something that exists in mathematics.
Brady, always asking the right questions, invisibly making his videos awesome.
+David de Kloet Brady acts as the audience surrogate, probably very deliberately. I'm sure many of the questions he asks are things he himself is wondering, but I'm sure he also asks questions he already knows the answer to, because he knows the viewers will have the same question. He's very good at this role. :)
I found it
I've found a truly remarkable proof for this, but it's too long to fit inside one comment.
😂
This joke has become less funny since KRplus removed the comment length restriction! Still worth it.
+Seth M-T I was about to make the same joke.
+Seth M-T
I have seen entire chapters of the lord of the rings posted in a single comment, though. :p
So... maybe in 2 or 3 you would have enough space?
+Seth M-T Fermat would be proud
A wild Ramanujan appears.
Quick, capture him before he dies of tuberculosis!!!
Joseph Crespo dude...uncool
@@josephcrespo7822 Or possibly hepatic amoebiasis.
Jacob R yeah i hadnt gotten that far, dude totally spoiled the ending for me!
@Jozef Wicks-Sharp That's just the phrasing Pokemon uses
This is probably the clearest video I’ve seen about the Riemann hypothesis. Most seem to focus on the setup and are kind of murky around the applications or the actual problem, but this one tackled the zeros very well.
Brady - when I subscribed to Numberphile 4 years ago, I wished to learn more about concepts exactly such as this. And for 4 years I've been blown away. The service that you provide; your great contribution to spreading knowledge... it's absolutely awe-inspiring. Thank you SO MUCH for making these (seemingly) obscure and complicated topics accessible to such a wide audience. I've commented this before, but I'll say it again: you are a great asset to this world. Thank you, and please never stop!
Hear hear.
Yes. Thank You!
There's a Numberphile 4? Where do I find it?
@@oli1181 I lold
There are infinitely many, but they're rare. Ah, mathematics... lol
I mean, throw a dart on a dartboard. The probability that the dart lands at exactly 0.00000...... cm from the center (aka exactly the center) is zero. Kind of wacky, but makes sense. Discrete numbers are an infinitesimally small infinite subset of the continuous spectrum of real/complex numbers.
kinda like saying: there are infinitely many integers that are evenly divisible by the first 80 billion prime numbers. there's infinitely many because the integers never end, but they're rare because there are massive gaps of integers that don't satisfy this.
Ok ok give me the proof that there are infinitely many.
great guy!
You don't need a proof. It's an axiom. We said there are infinitely many and so there are. Numbers are abstract.
8:30 multiplying out powers of 24 like that is definitely _not_ something most people can do at home - by hand. Ramanujan, what a gun mate
Yes! I'm always happy for more Riemann!
I'm also very glad you're expanding to L-Functions. The thought of something that ties the Riemann Hypothesis to Fermat's Last Theorem is pretty crazy...
8:31 Never before have I been so intimidated by the phrase "Just multiply this thing out"
Awesome video! Numberphile has really come a long way! Just 4 years ago I wouldn't have even dreamed of anyone daring to bring an advanced graduate-level math topic such as this one to a broad audience while keeping the mathematics honest.
This channel is the most fun a non-mathemetician can have with mathematics. Thank you!
"It's not likely, but it is possible." Thank you for the encouragement! :)
Possible in the sense of not being impossible. A small comfort.
I believe all those connections between the Riemann-Zeta-Function, Ramanujan and Fermats last theorem are the main reason I love mathematics...however, amazing video as always.
Same
RZH is truly the only reason I still care about pure math after college.
"There are infinite numbers of..."
"BUT YOU SAID THEY WERE RARE!!"
That genuinely made me laught. Brady sounds so betrayed. xD
"rare" has to do with their distribution. "infinite" means there is another of after each one you'd pick. the two are not mathematically, speaking, mutually exclusive.
I think it should have received a mention that there's actually another class of such functions for which the Riemann hypothesis has actually been *proven* around 1940. (Search "local zeta functions" or "Riemann hypothesis for curves over finite fields")
Riemann was so damn OG
+ZimoNitrome Fancy meeting you here, eh?
+ZimoNitrome Nice to see you here, friendo.
I see that you are also a man of culture.
one way to figure it out: make them puzzles in a game and put it on steam. players will definitely figure it out to the point where they take complete advantage of it
Or make a real life treasure hunt being the solution to this one of the steps, then post it on reddit and let it go viral
tatanpoker09 or offer a one million dollar prize! Wait...
LOL
Post on 4chan
@@royhe3154 That seems abstract for some reason. Now put it on steam and offer LIFETIME FREE GAMES AND DOWNLOADS?!?!?! Trust me, the hypothesis would be solved in less than a month lol
Ramanujan was the Mozart of mathematics in every single way. Such revolutionary people. Such a shame they died so young. They would have changed their respective fields to something we wouldn't recognise if they lived to a ripe old age.
I have to disagree. I think Euler was the Mozart. Ramanujan is John Lennon.
@@dhoyt902 this is tough but I would rate ramanujan slightly higher than Euler
Ramanujan wasn’t really all that smart, though, to be honest. He didn’t have any friends because he was such a nerd, and, because of that, he had all sorts of free time, so, sure, he figured out some obvious stuff while he was bored, but Euler was a true genius. He was doing calculations in the day time, and banging broads all night long. That’s why hardly anyone has ever even heard of Ramanijuana, and Euler is a hero and role model to literally everyone, everywhere. I mean, for reals, that ninja was such a legend, they even named an NFL team after him, called the Houston Eulers. Maybe when Ramanoodle gets a football team named after him he’ll be famous, too, but I wouldn’t hold my breath waiting for that to happen.
@@chriswebster24 😂 this is grade A trolling!!
@@chriswebster24 Clearly this noob hasn't heard of the LA Ramanujans aka LA Rams.
Video correction: Darwin did not develop his theory of evolution then go to the Galapagos. His visit inspired his theory :)
Thank you! A distinction mathematicians might, by the nature of their craft, be prone to miss. Darwin found a puzzle, sought a theorem, presented evidence, and stood prepared to be shown misguided. Maths developes by challenge-my-proof, and physical sciences develop by challenge-my-evidential-interpretation.
1:49 - ζ(-1) = -1/12 = 1 + 2 + 3 + 4 + ...
Brady, you're a sneaking fellow!
+TheHarboe Thank you for pointing that out. Never would have noticed it!
haha
hahaha, that made my day :)
More like Pete Mcpartlan did it.
but it's true :)
*youtube:* shows something I've never seen before
*My Brain:* yes, let's go
Brady, you are very good at asking questions
What I would give for Galois and Ramanujan to have had average lifespans... *sigh*
If it was possible, I'll happily give my remaining life to resurrect Ramanujan. I'm almost 19 btw.
He and many more Mathematicians and Scientists who died young deserve more lifespan(atleast the average lifespan) than a normal person like me :)
Lol!
Don't forget Niels Abel. Only living to 26 because of tuberculosis, the mathematician Hermite said, " He's left us with more than 500 years worth of math to figure out."
@@ericzeigler8669 you mean 32
100 subs before quarantine ends You are correct. Thanks.
I don’t have one degree in any maths, but this is one of my favorite channels.
Could you please do videos about all the Millennium problems? You have videos about Poincare and Riemann, and there's a video on Computerphile that talks a bit about P vs. NP, but I think that's all you have.
I support this idea.
I had no idea that Star Sapphire was into maths. Huh.
That's Scarlet Witch
As an engineer, I'm gonna give you a rough estimate and say it's true
And then a bridge collapses, killing 100 people
But you're happy because you won 1 million dollars
I have learned the hard way to leave the room whenever someone tries to explain anything Riemann related
I have learned to leave the video. Except that for some reason I don't. XD
And that's how your friends get you to leave parties.
Well, might as well Parker-Square it. right ?
😂😂😂
lol
Parker square is now a verb! This made my day
orochimarujes Groundbreaking result! Better keep it quiet so someone else doesn't publish it first...
Does anyone think there are 3D functions like this, with symmetric fields and planar zeros? Maybe finding these could help understand the 2D functions.
Why I like Numberphile: "There are infinitely many..." "But you said they were rare"
There is always a Ramanujan who came up with some crazy infinite series on any math topic
Really loved this video! Another great one for the Riemann Hypothesis.
8:31 - "This is something you can do at home" ... Yeah.. I think im gonna pass on that thanks!
What? You don't wanna have FUN with Mathematics!?
+Riotlight EZ use a computer to do it for you
x - 24 x^2 + 252 x^3 - 1472 x^4 + 4830 x^5 - 6048 x^6 - 16744 x^7 +
84480 x^8 - 113643 x^9 - 115920 x^10 + 534612 x^11 - 370944 x^12 -
577738 x^13 + 401856 x^14 + 1217160 x^15 + 987136 x^16 -
6905934 x^17 + 2727432 x^18 + 10661420 x^19 - 7109760 x^20
etc (You need a lot of terms to get it to work though)
Em how are you going to expand that stuff to the 24th power? Especially when its infinite... that'll take a while lol unless theres something im not aware of
If you take enough factors after some amount the coefficcients you get after expanding don't change anymore. For example, let's look at the product: x(1-x)(1-x^2)(1-x^3)... In every (1-x^n) the 1 basically states "Copy everything", and the x^n produces terms between x^(n+1) and x^(n+n-1), therefore if I would cut off the product at (1-x^3), I can be certain that all coefficients from terms upto x^3 in the final expansion are correct. The same goes for (1-x^n)^24, except that (1-x^n)^24 would produce terms between x^(n+1) and x^(24n+n-1).
It takes like 10s to expand the first 200 factors, guaranteeing everything up to x^200 to be correct. (It yields a polynomial of 482 401 terms of which only 200 have the correct coefficients though Lol)
Grade 8 distributive property.
"This is something you can do at home, just multiply this thing out." [points at an infinite product]. Uhh, sure, just give me infinite time ...
Incredible! One of the very best summaries of the RH that I've seen -- so succinct and helpful.
What a great video! Much more informative than most about the Riemann hypothesis
Thanks for this, in particular the Ramanujan example!
11:59 "you probably need to know some mathematics to understand that" LOL a bit of an understatement eh?
What a nice humble man - a great explainer
Srinivasa Ramanujan died way too young.
+Yiṣḥāq David So will I.
as did bernhard riemann at 39
His number was up
Yiṣḥāq David I believe we would be 50 years a head, at least in math and technology, if he had lived a full life.
John Von Neumann as well
Ramanujan is the king of the infinite series
@Nikhil Mankar such a shame for you
Why does Numberphile fascinate me considering that I don't even know my times tables and have never passed an exam in my life.
+Tom Mulligan Because real math is much more than what you learn in school.
+Shiny Rayquazza
Its a world away from school. I wish I had the capacity to understand more. I don't know what schools are like these days but if I were a teacher I would incorporate these YT videos into my class.
I’ve posted in this thread before, and once again I have to say how endearing this humility is to me. It seems to me that if there are infinitely many L-functions all having Reimann type hypotheses, and among them are the much lauded modular forms, then Riemann Hypothesis is not the answer. Rather, it is the observation of a symptom, and the cause is perhaps a deeper, most likely, much simpler idea. Following that line of reasoning, there should also exist a simpler proof of Fermat’s Last Theorem. In time I think the elegance and simplicity of this underlying principal will become known, and usher in perhaps the last great advance in our understanding of the object we call mathematics.
With that line of thinking you should be able to figure it out
Yep, Wormholes.
This is a pleasure to watch. Professor Keating's introduction is clear and informal. The graphics are also helpful, as are the series of questions and historical background.
Looking forward to your video on the recent proof claim!
Randomness chaos and predictably exist in all systems. It is the level of understanding of the system and in turn available information that determines how the system is perceived.
I lack the ability to comprehend what this man is talking about, but love and appreciate the questions Brady presented.
I dont understand anything, but i love this channel!
I see Wiles quietly and secretly working on the Riemann Hypothesis in his study ever since he finally fixed his FLT proof. Twenty years and counting. Wake up. sit at desk. Go for walks. Talk to the wife/kids. Go to bed and dream about it. Wake up and do the same FOREVER!
I'd like to think the average joe with a bit of math expertise might be able to crack it...but I don't think so. This is going to take a TRUE Mathematician with world-class skill.
I've been toying with it for about 5 years now, I got hooked on it after I bumped into the viral 1+2+3+4+... = -1/12 video. I'm a professional software engineer and a recreational mathematician. It's been a fun ride, mostly my "blow off" problem when I'm not dreaming up new ways to reinvent all of computer science with Homotopy Type Theory, Quantitative Type Theory, and a little something special I call Full Duality.
I've tricked myself into thinking I've proved it about once a year. Usually a bit of exploration or redoing the algebra demonstrates my error. I've been stuck on a few lemmas this year, some nasty limits that really look like they should work but just refuse to behave when actually doing the algebra. Maybe I'll nail it down one of these days.
Don't count us amateur mathematicians out, the crucial insight might come from not being exposed to the current methodologies.
Yep, my suspicion as well. He did the same thing for nearly a decade prior to his announcement that he solved FLT.
My favourite Numberphile video yet!
Like most "popular" expositions about the Riemann zeta function, this one has a HUGE gap almost from the very start: He begins with the definition of zeta(s) as the sum of a series, which converges only when the real part x of s = x + i y is greater than 1. Next, he states the property of symmetry across the line Real(s) = 1/2; but that makes utterly no sense unless and until one has defined what zeta(s) means when x < 0, and he has not done that.
i think it would be a great idea for a video to proof the euler formula for the zeta-function. It's not too complicated but really smart
Fun fact: Darwin didn't go to the Galapagos Islands to find evidence for evolution. He was a geologist as well and wanted to study the unique geology. He noticed that there were very similar finches etc... he came back with a theory, which he then started to study. (This may not be entirely correct, I seem to have forgotten some details. e.g. he may not have been a geologist, I just remember that it had something to do with the soil or lava or something like that)
He didn't have the theory yet until after he came back and started studying the specimens.
Numberphile evolves in the way it has to, I'm glad of this.
Nice seeing some math being done and not only explanations, and nice seeing modern math (not only Fibonacci or stupid debates about pi vs tau)
That was a wonderful and very clear explanation of the topic. Bravo!!!!!!!!!!!!!
Wow this Ramanujan was a boss
Mathematicians want to know the truth!
They will accept the truth!
This IS being humble to science.
Brilliant explanation, inspirational!
Thanks for a very nice description of the Zeta-f.
Erik Halvorseth i hope we get an answer
wait, let me get my calculator.
let me get my pen and notebook
Let me get my quantum computer
OMFG!!!! Why do I insist upon watching these videos when I HATE math...and I have absolutely NO idea about which they are speaking. Not even close!!!! Yet, I can't stay away..
This is my favorite type of Numberphile video! I'm very interested in the RZF!
I truly did enjoy your show
Nice .🎶😁
3:28 - It's interesting that it's known "at least 40%" of the zeros are on the symmetry line. How can we be so sure at least close to half of them are on the symmetry line? Is it because, though 10^36 is still well short of infinity, there is far less frequency of prime numbers at such large magnitudes?
+DAK4Blizzard Yes, I did not understand this part. Did he misspeak? Or is the emphasis in the wrong place? The animation does not clear up this mystery either.
+DAK4Blizzard It's possible that there has been a proof that at least 40% of them lie on the line - proving the Riemann hypothesis would be showing that 100% of them lie on the line, I guess.
+DAK4Blizzard I don't think that this has anything to do with the numbers already checked. What I guess is that you can estimate an integral which gives the ratio of numbers on the line to be at least 0.4. But I don't know.
+Elchi King (Maddemaddigger) Smart!
"we"?
Imagine Ramanujan living for at least to his fifties.
Thank you for this video !!
This is my favorite topic on this channel.
"There are infinite of them." "You said they were rare." Welcome to math.
WHOOOAA... The L-functions. Numberphile is getting serious.
+Peter Bočan From paperclips straight to L-functions and the Riemann Hypothesis XD.
very clear more of this guy pls
Amazing!! This the longest youtube video I couldn't understand a thing from. Well played.
Ever heard of VSauce?
Everything in his videos is simple... try string theory..........
I've spent years trying to wrap my head around what a proof of the Riemann Hypothesis would even look like, or how long it would have to be. One day...
Honestly, If I could, I would've traded some of my lifespan and given it to Ramanujan. I'm sure most will :)
Ramanujan would be imortal by then
A great channel, a very special channel for the 21st century
This was really cool, thank you!!
7:12 i read LOL LOL LOL LOL
There's no i in team but there is in the square root of -1
That just sounds a bit too communist. But there is an i in community. Also in society. Also in sociology. Hmm.
And according to my TI89 graphing calculator √-1 ≠ i. (Well unless you first do the variable assignment of *_i_* --> i.) Actually, it is a funny italics looking version of *_i_* but it is actually a different character than i. So it would be √-1 = *_i_* .
There's an i in team if the team is complex...
"There are infinitely many of them."
"But you said they are rare!"
How can anyone understand this, and not just be completely blown away? Mind = blown. Awesome. I so love math. Maths.
9:30 How did he got from the modular form to an L function?
How to make life more miserable for students?
"Hello, your today task on exam is very simple and easy. Find all zeroes of Riemann zeta function."
There are infinite zeros which make this problem impossible to solve. Also the paper would be too narrow to contain the answer.
@@hexa3389 woooooosh
That line at x=0.5, it almost feels like we need a third dimention to see how those points are distributed in said 3rd dimention.
Thank you!
For me, the z function looks a lot like an sphere surface projected to a plane. What if we put a second imaginary axis there and see what comes out of it?
Description has a rogue 'are' or missing 'to'
yeah it's pretty annoying now I've seen it.
ocd
It's a Parker Square of a description. :D
I can't even find it lol
Excellent video . Thanks a lot.
Fantastic video, thank you
Well this is well beyond my level..
"...that a non-mathematician could find this pattern..."
...I kinda feel like once you prove the Riemann hypothesis, you lose the right to call yourself a "non-mathematician."
I think the meaning there is "someone that isn't paid to do math and think about these things all the time." 150 years should have been plenty of time to find the answer if it was going to come from the standard process of iterating combinations of older theorems. As it has not produced an answer, I suspect we need some fresh insight, untainted by the standard mathematical mindset/worldview. That's what he was getting at, I think.
I wish Cliff was on every single Numberphile video!
Very well explained.
I have to study god dammit, and then you make a video on one of my favorite subjects
math is a game we don't know all the rule. so we play only in the know area. but will give it up for those crazy adventurers who will navigate the unknown sea.
Video is really Great! Trank-you very much
At 4:08 I think he means von Neumann and not allen turing. The turing machine was an abstraction, while the von Neumann machine was the actual implementation of it.
As a computer guy, I had to correct this
Turing helped build some of the first general purpose computers after using machines to help break the Engima encryption, as well as coming up with the hypothetical machine used in proofs. Von Neumann designed the basic components of the modern CPU architecture, yes, but Turing was also building real computers around the same time.
That's a "s=x+iy" formula
I'm so sorry...
s can be any number
y can be any number
x can be any number
but is has to be true the "i" is just extra part
+robin van Sint Annaland you dont get the joke?
sexy
You were the chosen one!
sixty
Nice -1/12 Easter egg. Almost missed it
I don't think this was stated in the video, but (correct me if I am wrong) that one generating function of x(1-x)^24... only has multiplicative coefficients if you look at terms whose degrees are relatively prime. For example, the coefficient on x^2 does not square to give you the coefficient on x^4.
I had to pause and audibly gasp at Euler's prime number transformation of the Zeta function.