What is the Riemann Hypothesis REALLY about?
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- Опубліковано 1 тра 2024
- Solve one equation and earn a million dollars! We will explorer the secrets behind the Riemann Hypothesis - the most famous open problem in mathematics - and what it would tell us about prime numbers.
I should have mentioned one additional property, namely zeros are mirrored along the line 1/2, even though non of them are found and maybe even non of them even exist. This way, every zero not on the line would giva a harmonic with Re(s) greater than 1/2, thereby breaking the estimates for the prime counting function.
Results discussed / references
List of the 7 million-dollar Millenium Problems: en.wikipedia.org/wiki/Millenn...
How to extend the Riemann zeta function: • But what is the Rieman...
Current best approximations for pi(x): www.sciencedirect.com/science... (Corollary 2)
"Implementation of Riemann’s Explicit Formula for Rational and Gaussian Primes in Sage": ism.uqam.ca/~ism/pdf/Hutama-sc...
"Some Calculations Related to Riemann's Prime Number Formula": www.ams.org/journals/mcom/197...
"The Riemann hypothesis is true up to 3*10^12": arxiv.org/pdf/2004.09765.pdf
Consequences of different zero-free regions on the growth of |pi(x)-li(x)|: A.E. Ingham: The Distribution of Prime Numbers, Cambridge University Press
Scene from Big Bang Theory: S12E6 The Imitation Perturbation - Наука та технологія
One mistake in this video: It is not true that Riemann knew the first 3 roots and made the hypothesis. He had over 600 zeros of this type, because he used what is known to us all as the Riemann-Siegel formula rediscovered really long after Riemann's death (the zeros are really difficult to compute without a computer, and 80 years after death of Riemann the world of mathematics knew less than 100 first nontrivial zeros. It was quite a shock to discover that Riemann himself had calculated many more than the rest of the world)
Just out of couriosity, how did they find out in modern times that Riemann himfelf new more solutions? Did they find one of his notebook?
@@XMarkxyz I am referring to the private papers and notes which Siegel studied almost 100 years ago. They must be in a museum or library now, I don't remember the details. These papers show an extremely skillful and diligent calculator instead of the intuitive genius that Riemann's papers suggest he would be.
@@XMarkxyz I mean, everybody knows the story that almost all of the private papers of Riemann were burned right after his death. They managed to salvage some few hundred sheets worth of material, which had never been a secret. They might have lied somewhere in the library archives in Goettingen, and many researchers had seen them before Siegel. Only Siegel could figure out what Riemann had actually done and how many roots of his Zeta he had calculated (and probably more since most of his private papers are lost forever)
It is also worth noting that we also know from Riemann's notes that his conjecture was not just based on extrapolating a pattern, but also because he was studying other zeta-function-esque functions, for which he was able to show that all of the zeros lie on a critical line. I forget the details, but it will all be in Music of the Primes by Marcus du Sautoy.
In college we were studying some insane calculation that Gauss made. Professor, we asked, how on earth did Gauss do all that number crunching without a computer. He had something better, she replied. Grad students. Seems the tradition of star professors getting credit for their student’s work isn’t exactly new. Of course, if you ever take a look at the list of Gauss’ students, they did ok for themselves.
“Ok guys, if this is the first time you heard of these ‘imaginary numbers’, let’s talk of a simple topic involving them: the Riemann Zeta Function”
That was a hell of a leap!
My mind exploded when you showed how the Riemann Conversion of the subtraction of the pole in s = 1 and the non-trivial zeros of the Riemann Zeta Function approached the distribution function of primes 🤯
This is the first video I’ve seen actually talking about the values of the zeros and showing them.
my mind didn't explode, thank god. I don't know how you made this comment if what you say is true
@@pyropulseIXXIyou dont keep a spare in a jar on your desk?
My mine exploded reading your comment! 😝 What's left is going to watch the video and finish the rest.
@@pyropulseIXXIBoltzmann might have something to say about this... xD
Holy. Shit. This video is CRIMINALLY underwatched.
Sharing it far and wide. I am a math phd (now in a different field) and, although I studied analysis, it is astounding that no one ever could explain to me, as well as you just did, how the Riemann Hypothesis actually matters to the study of prime numbers. Years of casual lectures and conversations. No one approached the explanation with your clarity. I have absolutely crazy respect for your ability to communicate this. Just. Wow.
Completely agree!
@@riggmeister Same!
As an EE familiar with signal theory:
You're fucking kidding me, it's just a transform? Basically a truncated Fourier approximation tweaked for the asymptotic behavior (the li(x) stuff), plus an error term -- and then taking the limit as n --> infty (for which the error goes to zero, or not, depending on proof)? And the zeroes are a kind of polynomial form of the transform of pi(x)?
And the transform has Gibbs phenomenon, just like my numerically transformed square waves?
That's so simple, surely it is wrong -- or else everyone else would use this as an explanation!??
True- but i have just clicked it after seeing the title :) - complex and quaternion analytics was sth that always attracted me. And even the german accent here is never a of a problem, hardly visible - thank you for the english pronounciation!
the harmonics are also part of number theory :) - the "perfect" numbers and partitioning - i really liked this video with graphs and images ! :)
Excellent video! I always found it kinda frustrating for math popularization that _the_ million-dollar question was not only so hard to explain (see the 3b1b video) but also even harder to understand why mathematicians care (I mean, you basically need an entire semester of analytic number theory to go through all the details of this connection between Riemann zeta and primes). Kudos to you for being able to boil it down with some incredible animations!
I agree that it's frustrating. My favourite alternative that people can understand much more easily is the good old Collatz Conjecture. That's always fun :)
Link to the 3b1b video?
@@macronencerلقد استطعت حل فرضيه كولاتز لكن كيف يمكن طرحها وضمان حقي في ذالك
For those wondering, the zeta function has a reflection formula such that the zeros in the critical strip have reflection symmetry across the critical line. i.e. say if s = 0.49 + 100i is a zero, then so is s = 0.51 + 100i. And it's that zero with the real part greater than 1/2 that would mess up that x^(1/2) error bound.
Very important gap fixed, thank you!
so you only need to search half of the critical strip
@@lolzhunter Technically, zeros of the zeta function come in 2 pairs, and you only need to search 1/4 of the critical strip.
@@RSLT sick
if you understand it why haven't you claimed the prize?
Explaining the basics of complex numbers and RH in one video. Man, you're a brave soul.
wtf is this comment? If you explain the basics of RH, then the basics of complex numbers is kindergarten stuff.
Your comment is the same thing as saying "Explaining the basics of addition and advanced differential equations in one video. Man, you're a brave soul."
@@pyropulseIXXI Imagine successfully explaining both to someone who knows neither, in one video.
@@Axacqk wow, that is actually amazing; I'm so stupid for not understanding your comment
@pyropulse7932 first time I am seeing someone on the internet man up to their mistake and learn from it. kudos to you!
@@agamkohli3888lmao
I've been looking for something like this for a while. I always wanted a Riemann hypothesis video that went a bit more deeply into the math. The concrete examples were really helpful too; like doing the error calculation for pi(10^50) or showing the sum of the first 200 harmonics. Great stuff
Completely Agreed, this is a great video! I cannot get enough of videos like these. I quickly subscribed and now browsing for more.
Also, if you like this, check out the ZetaMath channel. He also arrives at this meaning of how | pi(x) - li(x) | and 1/2 relate. But he takes you on a different fun journey of analytic number theory with lots of Euler and ending with how complex analytic continuation can help you find zeroes. Lots of details filled in. (and still going. the playlist is up to 5 videos so far).
I think you will love the series by Zetamath about analytic number theory and the Riemann Hypothesis, super interesting and clear and in-depth
Thank you for making this. This is the only video I have seen that actually explains HOW the Zeta function actually contributes to primes in how it is constructed. All the other have just said "this allows you to know more about primes", but this was really clear and informative, thank you.
This is the best video on Riemann hypothesis I've seen on YT. Congratulations on explaining it in-depth yet in simple terms.
Thank you for another excellent video! ^_^
If I may attempt to add something of value, Robin's Theorem is a relatively easy to understand statement about an inequality whose truth for all positive integers greater than 7 factorial (5040) is equivalent to the truth of the Riemann Hypothesis.
That is, sigma(n) < n * ln(ln(n)) * e^gamma
The inequality holds that the sum of the divisors of an integer, n, is less than the product of n with the natural logarithm of the natural logarithm of n, as well as e raised to the Euler-Mascheroni constant "gamma", with the 27 exceptions of 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, or 5040.
Robin's Theorem states that the Riemann Hypothesis is true if 5040 is the final exception, and is false if there are any more.
So, for those really interested in that $1,000,000, here is another way to approach it.
Riemann must have been an incredibly brilliant and intelligent man, to have seen all this with only 3 zeroes and no computers to work with. What an amazing genius
Was he even fully aware of how important his work would be relating to primes?
I always thought Euler's prime product formula was an ingenius insight which is fairly easy to understand (and forms the foundation for Reimann's work).
@@riggmeister Well Andrew, I wonder why you would say this? if anything. it makes it immediately obvious that you never read his seminal work on that subject as the title is : "On the Number of Primes Less Than a Given Magnitude" Anway, judging by the tilte alone, I would wager that he knew.
Riemann is considered by many objective mahematicians, to be one of the most influential mathematician of the 19th century, with parts of his work not fully understood by others far in to the 20th century.
He is a student of Carl Friedrich Gauss, a.k.a. the madman mathematician. It is no surprise Riemann was a brilliant mathematician.
He didn't see all this with only 3 zeroes. He postulated that all the zeroes lie on the same line as those 3 zeroes, which isn't really any insight, as it is just a guess.
But he didn't do this; he calculated over 600 zeroes before he made his postulate public.
Somehow, oafs like you think this reduces his insight, which makes no sense. Anyone could calculate 3 zeroes on a line and go "I think the rest exist on this line, too!"
@@pyropulseIXXI reported for fake and abuse. Watch your tone
I've seen many videos on this, and quite enjoyable they were too, but this is the first one that explains how all the bits fit together. Thank you!
Without the knowledge from ring theory, people will never understand the true deepness of primeness as a general notion.
Can you give a hint on how the understanding is deepened with the knowledge from ring theory?
@Number Cruncher They are alluding to the generalization of the Fundamental Theorem of Arithmetic (FTA). The FTA says that in the ring *Z* of integers, every number except 0, 1, and -1 can be written essentially uniquely as a product of primes numbers or their negatives, where "essentially" means up to order and up to the negative signs. In a ring _R_ , we call an element with a multiplicative inverse a "unit", so for example the units in *Z* are just 1 and -1.
When we generalize the FTA, it becomes a _definition_ rather than a theorem: we say that a ring _R_ is a "unique factorization domain" if - firstly, it is a "domain" (a certain kind of ring) and - if every non-zero non-unit element _a_ can be written essentially uniquely as a product of "prime elements":
_a = p_1 p_2 ... p_n_
where "essentially" means up to order and "up to units" - that is, if _a = p_1 p_2 ... p_n_ and _a = q_1 q_2 ... q_m_ then _n_ = _m_ and the _q_j_ 's can be relabelled so that for all _j_ , _p_j_ and _q_j_ are the same up to multiplication by a unit. "Prime elements" are analogous to prime numbers, and it turns out that in any unique factorization domain, the two possible definitions of primality of an element _p_ ,
1. its only factors are units (which are factors of everything) and unit-multiples of _p_ ,
2. if _p_ divides a product _ab_ then either _p_ divides _a_ or _p_ divides _b_ (or both),
both coincide - definition 2 is called being a "prime element" and definition 1 is called being an "irreducible element". In any domain every prime element is irreducible, but the converse is not true in general - the fact that the two notions are equivalent in unique factorization domains conveniently means that it doesn't matter whether we say "factors uniquely into prime elements" or "factors uniquely into irreducible elements".
@@schweinmachtbree1013 Note that it's better to say that UFDs are a classification of objects rather than a definition. We don't know, for example, how many things of the form Z[sqrtd] (all real numbers representatable as a+bsqrtd for integers a,b,d) are UFDs.
Any suggestions on a ring theory videos?🤯
@@iRReligious You're better off finding a textbook or taking a course. There's not much for higher level math on youtube.
Excellent. First video of yours I have seen. I am a retired electrical engineer that hung out with physicists my whole life. I find mathematics fascinating, luckily I have good math skills and I thought I knew a fair bit about the Riemann function and hypothesis. The stuff here is refreshing and new to me. Great job!!!
If you instead add simple periodic waves for all the nontrivial zeroes (a Fourier transform), you get a result that has sharp spikes at all the POWERS of primes, that is, all p^n where p is prime and n is a positive integer (plus a continuous component). It is very odd.
In the video, you can see small perturbations around the powers of primes as well (4, 8, and 9). No idea if that's related, but as you said, it's odd.
@@koenvandamme9409 Thanks to you and Matt for your observations -- that's a really, really interesting!
I'm very moronic when it comes to math but this was a joy to watch. I didn't fully grasp 5% of what you showed here but it made me want to understand more, seeing the prime frequencies emerge from the subtraction was absolutely beautiful, thank you!
Best video on the topic (and I've seen many). Amazing work!
Totally watchable, well done in showing how to present clearly a complex subject to anyone, thank you.
Great video. The first thing I've seen that does a good job of explaining why even a single zero off the critical line would be disastrous for results that depend on RH.
Thank you, this helped me 'understand' the Riemann Hypothesis much better than anything else I've encountered.
I´ve been watching some math related videos lately and they´ve re-awakened my interest and curiosity about mathematics, physics and other related stuff, since they´re mostly entertaining and fun to watch, while being very informative a sparkling. I´m a civil engineer and wish at least some of my professors in college were like these youtubers. Thanks!
A really wonderful and valuable video. So many videos about the Zeta function skip how one interprets zeros to determine the number of primes below a given value -- this one does not make that mistake. Great work!
I honestly have to thank you for this video as in no book or other video have I ever found such a clear explanation of what the whole endevour is about other than mentioning the fact that "if the RH is true, we'll know a lot about prime distribution"
This video is incredible. As an amateur math enthusiast (took nothing beyond ordinary differential equations), the mathematics behind the Riemann Hypothesis are well beyond me. This makes it much more approachable.
Amazing explanation, also great flow of ideas through the video. Also, nicely use of graphics ! Thanks for sharing this amazing knowledge with non-mathematicians! 🔥
This is the best Maths video I have seen on UA-cam. Well done.
Best and most accessible summary of the subject I have seen. Great graphics!
Fantastic video! I have watched multiple UA-cam videos on the Riemann hypothesis and this is the clearest and best one I've found at explaining precisely how the hypothesis relates to primes. Great job!
as a non-mathematician, I find it quite interesting and even mind-blowing. Thank you for your effort to present the material in an entertaining way.
Absolutely fantastic production quality! The sound, the animations, and… (what did I forget 🤔) Oh yeah! The *CONTENT* 🤣
Thank you for this gem! 🤗
This was very well presented, and honestly, I didn't think anybody would do it better than 3b1b, but you done did it...
Best explanation I found on YT. Great job!
Another nice connection between Riemann zeta-function and prime numbers is how the infinite sum of 1/n^s can be represented as an infinite product of 1/(1-1/p^s), where p goes over all prime numbers. Products of such kind are also known as Euler products.
4:25 is a bit misleading. This standard Mellin transform representation of the Riemann zeta function only converges for Re(s) > 1, like the standard series expression for zeta. Thus, saying that we are trying to find the roots of this representation is misleading since the zeros of zeta are all behind Re(s) = 1 in real part.
I really like the way you presented this. Mentioning Sheldon twice is a plus.
Fantastic video! Thanks for putting in the effort to make this :)
This is really one of the best explanation of RH, I need to watch it again and again, great job, many thanks and waiting for more.
another amazing video, the animations with the harmonics were incredibly didactic - not to mention pretty! This should be shown in classrooms!
Please replace “didactic” with a more appropriate word 🤦🏼♂️
The most accessible explanation I've ever seen (from someone that has a bit of maths). Congratulations and thank you.
Best video on this topic so far... thank you!
Incredible video. I now understand the importance of RH so much better. Thanks man
Thank you!
I've never seen this explained so well.
I have been looking for something like this for so long
Thanks! 🙏
After watching at least 5 videos, I finally have a better understanding of the connection of the zeta func. to the prime numbers, thank you!
Great work! This is the most amazing video about RH I have ever seen (i see about tens of video related to RH)! With reading the math bestseller Prime Obsession they are fully understand the meaning of the math problem and its beauty!
what a great goddamn video. So well edited too!
very well explained - first time i've understood any part of the Riemann Hypothesis!
Very good video. Number theory is fascinating stuff. A very good read on this topic is Prime Obsession, a book about the technical aspects of the Zeta function, as well as Riemann's life.
amazing video. the best video on the riemann hypothesis. I'm glad you didn't show it as a infinite series. I learned more from it that way. just a rlly good vid dude. idk I rlly enjoyed learning a bunch of new things
Very good video. Nevertheless, there is one very important omission: the Euler product, which relates the primes to the zeta function. Saying primes and the zeta function are not linked from the start is misleading. In fact Riemann, in his original 8 page paper on the subject, begins with this amazing mathematical relationship. By extending the variable s to include complex numbers he arrives at his extraordary results. Historically it is after this work that it started to be called the Riemann zeta function. So from the beginning primes and zeta are linked. Without a doubt Riemann would have not gotten very far without this deep connection discovered by Euler. Everything springs from masterfully manipulating this mathematical identity. Another thing is that he gave little importance to the what later became known as a famous hypothesis, he does not say it is such a thing. Riemann simply mentions in passing that maybe all the complex zeros are on the line but quickly moves on, basically saying it is not the aim of his paper to find that out.
It is indeed true, the link between the riemann harmonics and the prime counting function is the poles of the logarithmic derivative of the Riemann zeta function. This new function is computed using the Euler product to give the prime counting function and the poles are given by the zeros and pole of the Riemann zeta function.
Thank you so much for making this !
Man oh man...brilliant presentation.....please keep up!!
This was brilliant! Thank your for the great video
Very good job. You had to talk fast, but you got a great deal of information out pretty clearly. The graphics were necessary and first rate. Never boring and you held your direction well by not running down every complication, but not ignoring them either.
Legend... Thanks for putting this together
You deserve millions of views!!❤
Great video! I finally get how the zero's of the Zeta function relate to the prime numbers.
best video on youtube abou RH. Great visuals and explanation. Kudos to you sir
Fantastic explanation of concepts with a gentle guide to the symbols and now I am very curious to know more of the Reimann Converter...
Thanks for the video. It is both accessible and in-depth.
By far the best video on the RH on youtube! Thank you :)
7:54 I am a physicist and I don't think that the term "imaginary numbers" is misleading. Physicists use them a lot, but just because they make some computations easier. But they are never measured, unlike real numbers. Also even in quantum mechanics, where they appear in the wave function, one could use R² instead C to get rid of them as there is an isomorphism between R² and C. This isomorphism is used for example for all the diagrams of complex numbers used in this video (by identifying the real numbers with the x-axis and the imaginary numbers with the y-axis).
Would you say there's any value to conceptualizing of "imaginary numbers" as "scaffolding numbers" - i.e. they operate in the background and make significant calculations possible, but they're rarely of any direct use, and you typically want to stop thinking about them after the final product emerges?
this is a fantastic video! thanks so much for your effort. ❤
Omg. This the explanation that I have been looking for. Thank you.
This is amazing! Hands down best video on the topic I've seen (and that means better than 3b1b which is saying something!)
Amazing video! Always amazed by the beauties of mathematics
I've read Derbyshire's "Prime Obsession", which is about the RH multiple times, but you have insights here that I did not get from that excellent book. I'm really glad I checked this out!
Exactly my thought. Derbyshire didn't show that the build up of harmonics generates the steps at the precise values of the primes. Even though I have now seen it, it's still hard to believe.
Same experience.
Thanks for this video. It really helped me understand more about this problem - although still a lot I don't yet fully get!
Really nice video! I love it!
This is the best video on this topic !!!
I have a very loosely formed idea that has been kicking around my head the last few days related to this. It came about while I was playing around with the idea of a "unit circle" contained within only the positive real numbers x-axis and y-axis with an diameter of infinity (centered at 1/2♾️,1/2♾️) and hence an infinite circumference. This was mostly just a fun little mental lark for me into investigating the intersection of unity, infinity, zero, the infinitesimal, and their identity relationships, which then begin to branch into the possible relationships to primes and calculated precisions of pi when viewing the path along that circumference as the real number line starting from the points where the x or y coordinates equaled either 0 or infinity and calculating the arc lengths of sections of the that infinite circumference circle bounded by some whole number along either axis or working the other way from whole number arc lengths to where they fall on the axii. Since all of this has been primarily a thought experiment I began to get a bit into the weeds as far as the limits of my intuitive imagination so I need to begin working it out on paper to get a full picture and solidify some concepts I seem to be encountering, but the thought of the nontrivial zeros of the zeta function popped into my head unbidden several different times as looking like what an infinite circumference circle bounded arc length looks like when viewed with such a "unit circle", and I am starting to thing that if I take it seriously and take the time to work it out on paper and bring some other concepts into play like the complex numbers, natural log, etc. that I may be able to come up with at least an amateurish proof of why all the non-trivial zeros lay along the real part of 1/2 and that they indeed actually must neccessarily do so. Anyone think this is worth pursuing further, or no?
This is the best. Kudos!!
"This has to be the most impressive link of two seemingly unrelated objects in mathematics."
This is definitely good, but I think I still give the award to monstrous moonshine.
Great Job!!!! very Informative with well-explained easy method.
Best Zeta explainer yet. 👍
Absolutely stunning video.
best math video ever! amazing work! thanks!
What an amazing video! Thank you sooo much ❤❤🙏🙏
fantastic video, loved it
Another great video!
I literally had my mind blown. Good video.
I hope you don't mean "literally" literally 😆
Excellent video. I heard and read a million times that the zeta function had "something to do" with primes but watching the sum with the "Rieman converters" approach the prime distribution function was really my aha moment that brought everything together. One detail: the definition of the Riemann converter contains a function μ(n) that is not defined, unless I am missing something even after re-watching multiple times. What is μ(n)?
It’s actually not because they couldn’t choose another letter but, prime in Greek is spelled πρώτος.
Great video and content!!
A great explanation of the problem!
These are some of the crispest animations I have seen in my life, bravo.
Congratulation! excellent explanation
Is there a chance to find a derivation of your Riemann converter somewhere?
Tiny detail: imaginary numbers are not the same as complex numbers.
Imaginary numbers are multiples of i.
Complex numbers are combinations of a real and imaginary part.
I.e. imaginary numbers are complex numbers with real part zero.
Awesome explanation!
Hello. I'm interested to know, at 20:15, why did you choose 0.5+10i vs 0.65+10i? Is this an example of the discrepancy between the approximating function and the Riemann converter expression?
This is the first video to make me understand how everything is really connected.
Thank you, sir. Fantastic!
Excellent presentation.
Really liked how acessible the video is
"you certainly have the right to be mad at me for just claiming I exist without any explanation" 🤣
Amazing video. For those who speak spanish, Mates Mike has a very good video on RH as well, which I feel would complement this one very nicely
Great video! Very interesting
8:25 "C ≅ R[X] / (X² + 1)" should be with a *forward* slash (quotient of the polynomial ring by the principal maximal ideal (X² + 1)).
This deserves like a million views