The Riemann Hypothesis
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- Опубліковано 24 гру 2024
- The Riemann Hypothesis is one of the Millennium Prize Problems and has something to do with primes. What's that all about? Rather than another hand-wavy explanation, I've tried to put in some details here. Some grown-up maths follows.
More information: www.claymath.or...
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CORRECTION: The functional form of the zeta function is a reflection around the point 0.5, not the line x=0.5. But you can think of that as a reflection in the x-axis followed by a reflection in the line x=0.5.
For example, if we start with zeta(x + iy) and reflect it in the x-axis we get zeta(x - iy). Reflect that in the x=1/2 line we get (junk)*zeta(1 - x - iy). And reflect that in the x-axis again we get (junk)*zeta(1 - x + iy). When zeta(x + iy) = 0 so are all the others.
The zeros then look like this en.wikipedia.or... That's four zeros for the price of one.
The zeros are still symmetric around the x=1/2 line. The reflection of zeta(x + iy) is (junk)*zeta(1 - x + iy). In the video I mixed up zeta(1 - x + iy) with zeta(1 - x - iy).
But the point is the same, the error in the Prime Number Theorem is minimised if all the zeros lie on the line x=1/2.
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Some people have asked about my final claim that you don't get the prize for a counterexample. Here is the rule from the Clay Institute:
"if a counterexample is proposed, the SAB will consider this counterexample after publication and the same two-year waiting period as for a proposed solution will apply. If, in the opinion of the SAB, the counterexample effectively resolves the problem then the SAB may recommend the award of the Prize. If the counterexample shows that the original problem survives after reformulation or elimination of some special case, then the SAB may recommend that a small prize be awarded to the author. The money for this prize will not be taken from the Millennium Prize Problem fund, but from other CMI funds."
In other words, there are no guarantees either way.
The proof of the Riemann Hypothesis is clear and we leave as an exercise to the reader.
andersjaevel damn that’s good one lmao
You need a couple of pints and few less text books mate.
Some good Landau right there.
Trivial*
@supadox Because often, the proof is rather easy to make if you did understand exactly the proofs and methods the textbook used to prove some other thing just before.
And it is actually a good excercise to come up with ones own proofs. Though it is extremely hard i think.
This is a great explanation of the Riemann hypothesis. I like how you presented it in historical context, broke it down into parts, and explained its relationship to other aspects of mathematics. Very critically, you presented the equations, you explained the intuitive portions to non-mathematicians, and you did it with the enthusiasm that is difficult to glean from dry textbooks. Love your UA-cam posts and your appearances on Brady's videos. Keep up the great work.
+72ackerman Thank you very much!
+singingbanana Also, you finally taught me what that big Pi means before an equation, which I had not known before! (Math looks less like hieroglyphics now).
+72ackerman Ha, I'm glad.
+72ackerman
The symbol
is used for continued fractions. Well, not really, but I like the idea of using actual hieroglyphics for math.
+christosvoskresye And, of course, it doesn't show.
"It was like the mathematical equivalent of a mic drop. Peace out, that's it. I'm done" ahahaha I love you Dr Grime
I believe that we just got -1/12'd again.
I find it interesting that I watched this video about 2 years ago when I was 17, and I remember I didn't understand somethings. Now I'm on college, and I decided to watch this again, and surprisingly enough, I got a lot more of what he said. It's nice to see our own development in practice.
I am looking forward to understanding this:)
I feel exactly the same way. Although I failed both of my Math classes, I learned quite a lot.
+Dhiego Bersan Same here except I was 14 when I first watched this and now I'm 16 and I understand everything perfectly.
+tggt00 bahahahahahah
What's so funny?
As a pure mathematics major, all I want is to sit down with you and learn even more. Wonderful video as always, Dr. Grime.
James, sincere thanks for following through on this request by interested viewers like me, I am familiar with the content but the nuances continue to grow over time. The internet access to information sources like these youtube channels is a stunning contrast to when I was first introduced to it reading a paper back copy of "Stalking the Riemann Hypothesis by Dan Rockmore" around 2005 "by candlelight". Watching this now three or four times over two days was again similarly juxtaposed personally.." very exciting" and a thoroughly helpful addition to the initial "incredible read" experience. The interval between then and now is remarkably influenced and is clearly given me access to real structure by better understanding the mathematics, especially in the margins of addition and multiplication. The theorem's wide appeal to many pursuits is in itself remarkable. In all the video here is "Exquisitely arranged material and lucidly presented with admirable understanding and enthusiasm in a skilled detailed explanation". So much of what excites me about mathematics is how much there is to learn and the many open questions still abound in this wild field... exploring the unknown... and its good to leave a trail!
Whenever I feel too clever, I just tune in on this channel and I feel stupid again. I love how passionate Jim feels towards math. Makes me passionate about it, even though I'm no strong mathematician. Keep it up :)
It is brilliant. Clear explanation of Reimann Hypothesis is given, and it is inspiring.
I love to eat my dinners and watch this dude. His videos are usually long enough for me to finish my food.
Too bad I don't understand a single thing he's saying.
I love that I'm somehow a strange part of your daily rituals. I have a similar habit.
There's a Japanese club for families to learn a bunch of languages, and the funny thing is that the guy who started the club requires people who join, as their first assignment, read Heisenberg's book on quantum theory.
He says that it won't make any sense to you, but that if you're going to learn new languages you're going to have to get used to reading things that don't make sense to you...
Though, obviously, some of his members did come to understand Heisenberg's book because they published a book that goes through every single step of proving that Schrodinger's wave equations are equivalent to Heisenberg's version. It was written by students who had just learned it, on the theory that you're better at remembering all the details and explaining something that you've just learned. And it's very Japanese, full of cute little cartoons and asides like how to draw a cartoon the physicist De Broglie, who's face is shaped like a peanut.
Popexssj same here bro. :)
Aaaand, it might be easier to earn a million dollars selling hotdogs :D
it all depends on how you go about it ;)
Felipe Budinich Or just borrow a small loan of a million dollars
Deepto Chatterjee And you’d have to pay it back within a long time interval
Just rob a bank, or build one!
If I came to the UK could I take you out to lunch or something? Through your fantastic presentation style (and I really mean you, not just numberphile) you've managed to get my mathematically agnostic SO presenting proofs to strangers at restaurants. Between this feat of magic and your continued ability to inspire my own insterest in maths beyond a lowly undergraduate minor, you've quickly become one of my favorite people on the Internet. If not lunch, please let me know some other way I might support you!
You, sir, are a gentleman and the gratest math comunicator ever. Thank you.
Dr James Grime, I salute you for finally making this video! :)
The natural length of this video turned out to be 20 minutes. Normally they're only about 6 minutes. That proves something, but I'm not quite sure what yet.
singingbanana New proposition: Any attempt to explain a currently unsolved mathematical proposition/hypothesis will take about π times the expected length of time.
singingbanana It means this video should've been a three-parter.
Great lecture. Your deep knowledge is apparent, but you're also an excellent communicator. Thanks for the videos you've made.
You take those topics which were always presented to me in the driest possible ways while I was working on my Maths degree and make them fascinating.
I can't believe I just found this channel. I can't believe this is the channel name. I can't believe you have Little Professor as your banner.
I'm very pleased.
CARITO BENAVIDES MONTILLA
MiUltimaHistoria
Pensaba yo en estos tiempos difíciles
Llenos de incompresiones y desencantos
en tantas buenas razones Para soñar
Y sin quejas, sin desfallecer, si… con todo
El agrado y los buenos sentimientos…
pensaba con nostalgia
De la inocencia de aquellos años
De esos primeros amores
Con yannette, teressitta, floralba…
Si… florssitta amore mio,
Que al recordarlas, vuelven a nacer
Y recuerdo en mi memoria
Aquellos tiempos de … de Carito
CaritoBenavidesMontilla diMugnoz
Mi bello, nuevo y gran amor….
Ella era una miss de Boston
Ocasellb: OcassoDiBellSetBuck
Que daba clase en la escuela di
Cullinnarii et pannes también…
Le gustaba el español
Y aunque lo hablaba poquito
Tenía esos ojos bonitos
Que hablaban muy bien por ella
Carito me habla en inglés. Aveces…
Que bonito se veía sola o en compañía
De unos que admiraban como io
Lo que jamas seria per me, perque
Io ya era cassado: con Florsitta, Con DianaCarolinna, y por supuesto, conLaFuncion Zetta di Riemann
Carito me habla en inglés et también en Italiano, germanny, francais, portuggisse,
Mandarin pictórico y por su puesto, En un quichua, básico para viajar por ecuador, peru, Bolivia, chile, y volver al final de tanto andar, de tanto amar,
de tanto padecer su belleza,
Que me dice: yo no se
Carlitos don´t be like that
Now listen to me
You will pay attention
I need you to write in english
Muy perfecto parragraph
And tell me where did you learn
Donde tu aprender to be tan coqueto
Remember nada de futbol
Until you finish the work you have
Y me daba una sonrisa
Y yo me quedaba loquito
Y despues en el examen
Lo ponía todo al reves
Carito se fue del valle Yo la recuerdo cantando Porque me dejo muy triste cuando Para su tierra se fue
Carlito dime que si, me eneñaras todo
Para no quierer morir per te…….
Carito don´t tell me no
Que me muero por tu amor
Que importa la raza
Tampoco el idioma
Si al fin lo que cuenta
Es lo buena persona
Si es del altiplano
O de tierra caliente
Si al fin lo que vale
Es que sea buena gente
Que importa su credo
Si es hombre de influyente
Si al fin lo que cuenta
Es la gente decente
No importa si es blanco
Si es pobre o famoso
Si al fin lo que vale
Es que cante sabroso
Pudiera escribir un millón de tequieros
Y algunos teamosrepletos de sinceridad…
Pero esta vez, ire lo mas lento posible…
Paso a paso, dia a dia, verso a verso,
Para que ni tu ni yo se pierdan
En tanta vanalidad en que habre envuelto
Mi extensa vida que alguien resumirá:
Could you do a video on each of the clay millennium problems?
This is by far the easiest of the six to understand. The rest of them, particularly P vs NP, are so abstract, just the formulae used to write them require a PhD in migraine medication.
@@cosumel i know i'm late as you can possibly be, but anyways.
P vs NP is by far the easiest to explain. You can even explain it to a child. Now try that with the RH.
It gets abstract and complicated when you start working on it, basically like all the other Millennium Problems.
This guy makes even the most daunting subjects seem approachable. It's so much easier to stay excited and learn like this. I wish all teacher were like this, maybe then more people would be interested in maths (perhaps then, people who are naturally drawn to maths and physics wouldn't feel so isolated *sighs*).
"..that Riemann decided to rip off, I mean study." I coughed up my chocolate milk.
The primes is a vortex number that goes back and forth with the number (0) in the center 7+7+7 opposite 3-3-3-3 also 6+6+6+6 opposite 4-4-4-4 also 9+9+9+opposite 1-1-1-1 in the case of the number (5) it doubles on it
I dropped out of grade school and almost got this! It got hard after 0:08 though.
It is not enough at 15:10 to point to the function Li defined at 3:45 for real numbers, because now you need a function that is defined also for complex number x^ρ, where ρ is a nontrivial zero.
I realize this video is old. But is there a follow-up video which explains more in detail how to handle the function Li defined on a domain that includes non-real numbers?
I studied math for a while a couple of years ago and I almost understood all of this. I'm so proud of my brain.
same feeling XD
By far the most understandable and interesting explanation of the Riemann Hypothesis.
Thank for the crash course, James.
I've read cash course at first
The 5 travels down the grouping of 6 from the 5th position to the first position to the 25 and cancels out the right prime to 24.
The 7 travels to the right through the first position back to the first position of 48 and cancels out 49.
The squares of primes cancel out other primes. That is why the number of prime numbers less than a certain number follows the inverse of the log
I feel ridiculously out of my depth watching this video. I only feel as though I partially understood this because you led us through it in such a logical and sequential order! Thank you for the brain strain! :)
This one isn't for everyone. I'll do something more accessible next time.
I know this post is old, but don't feel too badly IMO. I have read a book on this, and studied a lot of math in college, and it's not simple to follow. I think I would have been equally lost had I not read a book on this, and remembered the equations. I was confused a few times by the book ( Derbyshire,) and this video is actually simpler, but it's dense. I would benefit from watching it again, I think.
@@singingbanana The Riemann hypothesis is a distraction
The True lie on the Sum of all Natural Numbers
The answer to this problem of -1/12 is that Jesus come back to say to everyone "i have 12 Apostles"
I can definitely see why people ask you to explain that one. Sucks about no million for disproving the hypothesis. Great return to posting videos! Those formulas hurt my eyes and that's awesome.
the person who proves it, if he/she has watched this video should go back and leave a comment like: "lol, proved it."
yea gonna do that, just give me some time
Looks like someone did it
I transferred from a community college and began my first quarter in an actual university in California (UC Davis). I transferred to study math but had no interest in it at all, it was just a way to move on. I wish I began watching your videos earlier, here and on numberphile. They've made some hard to grasp topics much more enjoyable and understandable, such as the video where you deal with the different types of infinity. I look forward to newer videos from you and everyone on numberphile.
When you say log(x) are you referring to log_e(x)? It seems no one wants to keep consistent notation with it.
Also, I'm happy you finally did a more intense video, but I find it sort of funny how people are acting like it was a super difficult video when you glossed over all the actual difficult details.
Even the residue calculations can get horrendous.
I sympathise, even the statement is more advance than most people are familiar with. It is the natural log.
14:10 Does anyone know a reference for the equation ζ(s) = π^(s/2) /(2 (s-1) Γ(s/2-1) )∏ρ (1 - s/ρ)?
Is it possible that it is true but there is no way of proving it? Or if it is true can we the be sure that it is also possible to prove it?
I'm really happy to see a new video on your channel. Numberphile doesn't always go into the same depth as you do on this channel.
i have no idea what i just watched but i still enjoyed it.
A Solution for the RIEMANN ZETA FUNCTION is extremely valuable because It also point to Solutions for enhancing the HAMILTON GEOMETRZATION Poincare conjecture, Hodge Invariance conjecture as it relates to PRIME NUMBERS, TAYLOR INFINITE SERIES ,MACULIN SERIES and Doing Arithmetic past ZERO or Singularity as it is called in Analytic Geometry , and Algebraic Geometry, and it Directly points to the Prime factorization Algorithm , the Division algorithm, and the QUADRIATIC FORMULA This Solves many DIMENSIONS and RANK IN THE COMPLEX FUNCTION PLANE for MANIFOLD like The Kahler MANIFOLD ,CALIBU YAU MANIFOLD simeoustanesly and Points to Soulutions to the entire Millennium Prize Problems proposed by The Early 20th Century Philospher and Mathematician David HILBERT , Including the YANG-MILL Mass GAP , and the NP COMPUTATION time space COMPLEXITY problem also know as the Traveling Salesman problem
Thank you, been wondering how complex numbers were needed to do PNT proof, am hoping to learn more about this material
I cannot state having understood, but I'm greatly inspired to learn more to understand it finally. Thanks James and don't let the gap between consecutive videos be as big as those between big prime numbers! ;-)
the number (0) is on the same line because it is a number guard it grows to forget and to the right and the center is still, just observe in the sieve of eratosthenes
I do not have the education necessary to watch this.
Sad thing is, I have the education necessary to watch this, but I don't have the patience to listen to him. I feel like I'm listening to a lecture from a professor asking myself the same question I did then "Why the hell do I even care about this?". No offense to the presenter, he is a great guy and math is his thing and I'm glad he can get excited about it. He's also made some other cool videos that have kept me entertained, but this one is for the egghead nerds for sure.
GameRage Dad I'm the opposite. I have no idea what he's talking about but I love listening to it nonetheless. Most of the reason may be his adorable, awesome accent.
khaled khunaifer You're hurting my brain.
khaled khunaifer yeah... you missed the complex analysis part, we all wish we can LOL
***** Seventeen with the math ability of an eight-year-old.
I am currently doing a doctoral degree in Physics where I use the Riemann zeta function all the time. I had no idea of its origin or how interesting it really was, just kinda used it as a tool up until now. Great video, really enhanced my interest on the subject!
That's awesome - thanks!
9:20 - This isn't strictly true, since he didn't explain the restriction that the function must be analytic.
Oh, sorry:( I hastily hit a thumbs down on you're comment w/o thinking. You are of course completely right.
This video summarizes in 20 minutes all odd-numbered chapters of "Prime Obsession". I liked!
Looking at graphs of Riemann Zeta, would it be wrong to infer there are some fractal properties of primes?
Probably, but you should rephrase that to prime properties of fractals. Primes show up in almost every aspect of math; at this point it would be more impressive to prove something is not related to the primes.
Goldbach's conjecture works because all odds are even with (1) more and the primes are odd only divided by (1) summed the primes sum (1)(1) which is even
Are you aware that there is a direct correlation between your enthusiasm and emotions attending this explanation and the actual rigor used in the ramping up of your description? It's almost melodious.
What if numbers and math have human emotive and fantastical elements to them? What if math is an endless song or a dance that we can't sing or sway to because we lack the dimensional articulation? Thus it behooves us to merely express things mathematically using one-dimensional verbal or written symbolic gesturing.
This idea is only partially reflected in the quiet passion of too few mathematicians or scientists.
John Nash understood it.
***** Sounds good to me.
+Peter McMillan Agreed. Especially where he spoke about proving that the zeta function has no zeros on the line of the real part of S = 1. Hurrah.
Likewise from the WIKI, "In a lecture on prime numbers for a general audience, Fields medalist Terence Tao described one approach to proving the prime number theorem in poetic terms: listening to the "music" of the primes. We start with a "sound wave" that is "noisy" at the prime numbers and silent at other numbers; this is the von Mangoldt function. Then we analyze its notes or frequencies by subjecting it to a process akin to Fourier transform; this is the Mellin transform. The next and most difficult step is to prove that certain "notes" cannot occur in this music. This exclusion of certain notes leads to the statement of the prime number theorem. According to Tao, this proof yields much deeper insights into the distribution of the primes than the "elementary" proofs."
Maths are the tools, but music is the spirt.
+doceigen chaos theory proves the relation between you and me, music and me, mathematics and me, using mathematics.
Thanks for making this video. I had the Riemann Hypothesis explained by a professor once, (in a most excellent way too, outside with sidewalk chalk)... but I forgot much of the explanation. I'll have to watch this a few more times to really have it all sink in. I'd like to see more videos like this with the right level of detail without getting bogged down in the formulas too much. I especially like pointing out the important parts of involved formulas.
Is it a british-ism that you call 1/3 "a cubed" instead of a "a third"? The other ones seem the same as North American usage. He does this twice around 5.40.
I'm British and it's not something I recognise. I'd even say he misspoke, but it could be a regional thing.
You sir have a talent for teaching and for communicating information the best way possible. I thank you with all my heart for these explanations and these videos. Keep up the good work!
Don't tempt me to switch my major to mathematics.
***** Perfect reply :)
I Wish I had the opportunity to do so when it was my time!
Is a great explanation of a non trivial topic, a wide background is needed but is understood with your guide! Great work! Thank you
I love this guy. Always very clear. Great video.
Ok, it took me a few minutes of goggling to realize you're not "Numberphile", just a part of it! But let me tell you, you MAKE Numberphile what it is! Love your enthusiasm and videos!
Why is the English notation off the natural log just 'log'? Where I live, 'log' means log base 10 while 'ln' is used for the natural log.
I usually see “ln” in English, but I guess “log” just looks nicer and it's fairly obvious that we're not dealing with base ten logs.
It depends on whether you're an engineer or a mathematician, the former uses ln for natural log, the latter, log and they say log 10 when they mean log 10.
Because the natural log is more important in maths and log10 is better for applications. You are right that log should mean log 10 but log 10 is not very useful in math theory so log tends to mean natural log which is much more useful.
Exactly what Ryan said. Mathematicians always mean the natural log.
singingbanana Jeroen J Exactly. And Jeroen, presuming you're from the Netherlands as well; I study mathematics at the Utrecht University and we use log as in base e, but in high school I was taught log was base 10 and ln was base e.
strikingly beautiful, amazing clarity, please do more
Picturing Riemann doing a mic drop and saying "Peace out, y'all" had me dying.
the fact that i dont understand what you are saying makes my depression come back. I genuinely just want to know what mathematicians are talking about but i think as soon as i graduated high school i lost that skill
great to see you back with a video :D
You did an awesome job explaining this. I have been reading/watching videos on this for the last couple days trying to find a sufficient explanation of what in the world the connection is between the zeta function and prime numbers, and you explained it well enough that I am sufficiently satisfied. Thank you!
Am I correct to assume that those symmetrical non-trivial zeros that you plotted do not in fact correspond to any actually known zeros because otherwise the whole Riemann hypothesis would have been disproven already? Isn't it a bit confusing then to just put them there?
Yes, you are correct. I guess he was just illustrating the fact that IF they were to exist as shown, they would have to be symmetrical about the ½ line. I agree that it may have been a little confusing to some people and he could have made it clear that he was showing hypothetical zeros. But a good video generally.
At that point in the story, we had only achieved that information. It was only later with the Zeta function that we got the idea of them beeing on a single line. Furthermore, that proves a point. Never trust a mathematician, and never ever trust a picture! :)
so non trivial zeros when we change s as complex number z=1/2+it in midlle equation blue time 12;00 ?
there's a very interesting book "Prime Obsession" by John Derbyshire, it is all about primes, zeta function, the riemann hypothesis, and Riemann himself :D
yeah very interesting book, it also talks about the connection between HR and physics
It also has the wildcard numbers (2) which is the only pair that goes through every pair also the (5) that doubles itself 5.10.15.20.25....
10 years later, I'm still waiting for the proof that cat equals dog and dog equals cat
Thank you! I spend a lot of time speaking with my classmates (we are studing maths) about the series 1+2+3+... after the numberphile's video. I refuse their 2 "proof" (because I lerned in analysis 1 that 1-1+1-1+... doesn't converge, so it isn't equals to 1/2 and in the second video they set -1 in a formula (coming from a geometric series) where |x|
fine ill do it, hold my beer.
i thought you drank soda
Done yet?
yo, this beer is getting kinda heavy... or at least the glass is.
30 years later... "Hey father, why are you always holding the beer?" "He's still solving it... Oh my dear"
Surely OP will deliver
Good job my brother. If you are still there,thank you. I never understood riemann until today. You made it simple
in tenth grade atm, almost got the jist of it, hoping I'll be able to understand better as I learn.
So can someone confirm that I've understood this correctly, the riemann hypothesis is equivalent to the conjecture that if the blue equation (at 11:54) is equal to zero it implies that the real part of the input is equal to 1/2?
i thought not the non trivial zeros when real 1/2 + it .we get than s= z complex number ,but the non trivial zeros from a Riemann siegel formula and not the blue equation at 11:54 i wait for others to confirm this answer ?
19:17 That's... Lame, I mean I get why, you want to pay the guy for showing up, shake some hands and say some words and being a symbol for the problem. But still that's not how math works, solving the problem means either proving it, disproving it or proving it's "improvability".
If you disprove it they'll be bummed out
I think the best part of this video is he explicitly defined the Riemann zeta function across the entire domain. Most authors just give the infinite series definition and say “by analytic continuation” everywhere else. Which is totally useless
Every time I watch this, I understand it a little bit more.
2 questions. 1) is there a web site about this. 2) if the euler function only matches the function for s>1, then how can the extension of the zeta function help show more about primes?
The harmonic series sounds nice, at least the first few terms. The seventh is a bit flat, though.
If someone still reads this: it is often said, if we could prove the Rieman Hypothesis, it would give great insight into the nature of primes. Is that the case though? We already have that insight, we just can't prove it's the correct insight. Proving it just means: "yes, what you now only assume is true about the nature of primes, definitely is true". In a way that's how every other science than math operates all the time. Just assuming.
Mathematician here... it's the journey you take when u prove the statement that gives great insight.
To prove this theorem... u literally need to take a hammer and break open numbers and peer inside them. See all the structure inside them. And inside everynumber... even pi... all that u will find inside them is primes.
Prove RH ... then u will know how primes work. U will know how God works.
God made the primes... that's it.
Even though I have almost no idea what he was talking about, I still feel a little bit smarter just watching it xD
Great Video. Very well explained the R.H. principles. Fantastic Job!
What if...the answer to Riemann Hypothesis is the friendship we found along the way?
I'm learning about infinite sums at university nowadays im learning chemical engeneering at Budapest. But i just got in love with this kind of maths and i was amazed by riemanns state about the Conditionally convergent sums ! i wish i could count better with complex numbers to do tricks like he did with the dzeta function :D i mean thats amazing that you can get the sums of divergent series i mean if the dzeta function is extended by this way and it has such a huge impact on the real numberline and the prime numbers then i'd say that the sums that can be got out of divergent series can be really useful in other sciences :D!
Why am I even watching this? I understand about as much about math as a banana! ... *continues watching* ....
Hello, I watched the whole thing and I think I fairly understood the behaviour or the function and the relationship to the primes and the prime number theorem, and the proof would be showing that all zeros have to lay in the +0.5 line and non can exist in any other place. I have heard (or maybe seen in a few TV shows and movies) that proving Riemann's Hypothesis could have security implications since internet security is heavily based on prime numbers operation. My question is, what result could that proof yield that would be more significant than the results Riemann did making the hypothesis? Maybe a specific formula for all the non trivial zeros? Didn't he did that already? Could someone just assume it is true, use the final results and work something out? Or was Hollywood messing with my head again? I mean Riemann already worked out the consequences of it being that way, are there more consequences for knowing for sure it IS that way?
I really think it's pretty crappy that you don't get the prize for disproving it. Wouldn't it be funny if someone disproved it and kept it to themself since there wasn't an incentive to share? Of course, that's impossible because anyone that dedicated to mathematics wouldn't be able to keep it a secret.
You'd still be famous.
Oh yeah, I forgot to tell everyone, I disproved this years ago. It was so disappointing that I burned down my office with all my research.
I'm not 100% sure if he just made this part up or not, or where I could search to get a reference to that fact.
I think you would become the most loved and at the same time most hated person in maths.
I guess he meant disapproving with a rigorous proof will get you price. Disputing with a counter example won't.
When I saw this was about the Riemann Hypothesis, I wondered if the 1+2+3+4+5+6+... = -1/12 thing would find its way in here as well, and was not disappointed.
Why log base 10?
+Cyrusislikeawsome That's natural log. At this level, log always means natural log.
+singingbanana Ah, thought so, thanks. Its just that my maths teacher/calculator usually says log on its own refers to 10 and I'm more used to seeing ln. THANKS! Love the videos!
+singingbanana Wouldn’t it be better to have the same symbol in every part of mathematics?
+singingbanana I thought natural log is base e?
David Alexander It is
Dr James, I think I got a better chance winning the lotto than to win the mathematics millennium prize. I thought I had a shot in solving this hypothesis.
11:23 -1/12?!
NOT AGAIN!!!!!
lolzomgz1337 the curse of numberphile
..I'm a bit confused. In Numberfile video on this subject ( Million Dollar Math Problem - Numberphile ) at 14:18 prof Ed Frenkel says that disproffing the RH will winn you prize. It' makes sence that if you find the counter example of the one zero that doesn't lie on critical line means that all zeroes (except obvious ones) don't lie on the critical line, which is contrary to the RH. In that case RH could not be proven....:/
They paying one million dollar for this proof is like me paying 0,15$ for Lobster/Champagne dinner.
Even though I didn't understand this enough to understand the Riemann Hypothesis, I still learned a lot from it. More of this so-called 'maths' stuff, please! :)
That x = 0.5 line is called in some circles 'the dirty crease.' Modern maths seeks to know how dirty the crease really is. Get at it.
I've always wanted to know what the Riemann Hypothesis was about.
Thanks for making it a bit more accessible because wikipedia sure isn't helpful!
Thanks. I think the choice is a hand-waving explanation or a difficult explanation.
Fermat be like: i know the proof. But this margin is too small to write it down
A Solution for the RIEMANN ZETA FUNCTION is extremely valuable because It also point to Solutions for enhancing the HAMILTON GEOMETRZATION Poincare conjecture, Hodge Invariance conjecture as it relates to PRIME NUMBERS, TAYLOR INFINITE SERIES ,MACULIN SERIES and Doing Arithmetic past ZERO or Singularity as it is called in Analytic Geometry , and Algebraic Geometry, and it Directly points to the Prime factorization Algorithm , the Division algorithm, and the QUADRIATIC FORMULA This Solves many DIMENSIONS and RANK IN THE COMPLEX FUNCTION PLANE for MANIFOLD like The Kahler MANIFOLD ,CALIBU YAU MANIFOLD simeoustanesly and Points to Soulutions to the entire Millennium Prize Problems proposed by The Early 20th Century Philospher and Mathematician David HILBERT , Including the YANG-MILL Mass GAP , and the NP COMPUTATION time space COMPLEXITY problem also know as the Traveling Salesman problem
Excellent lecture - it brings light into a dark place. Quite enjoyable!
I'm a physics student, I have a lot of math as well and I really like math, but I didn't get any of this :P I still really liked the video. James can really make such a thing really enjoyable!
if Sum n, n = 1, 2, 3, ... = -1/12, how do we use this result to derive further interesting results?
Sadly been too long since I did maths like this. I'll leave the proof to you :)
This is certainly the most insightful video I've found on the properties of the Riemann Zeta function. I'd just wished you could go more slowly over some of the steps, but they can be filled in with some effort. Great job!
Yay youre making videos again.