“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!
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 :)
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).
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.
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."
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.
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.
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)
@@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.
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!!!
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!
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.
I've accidentally proven that li(x) and pi(x) are inseparable. I don't know what to do about it, I don't want to be called a crank or a crackpot, it was an accident. If there's an expert in this problem that is interested, I can talk you through my process, maybe you could confirm this for me? It would be easier than multiple rejections, I don't have credentials, instead I have extreme imposter syndrome and a solution to this problem. I don't know what to do. I really could use some help and advice. Also, it's extremely simple, but it is time consuming.
I've accidentally proven that li(x) and pi(x) are inseparable. I don't know what to do about it, I don't want to be called a crank or a crackpot, it was an accident. If there's an expert in this problem that is interested, I can talk you through my process, maybe you could confirm this for me? It would be easier than multiple rejections, I don't have credentials, instead I have extreme imposter syndrome and a solution to this problem. I don't know what to do. I really could use some help and advice.
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.
Fantastic video! I have watched multiple TH-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!
@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 1 and _p_ , up to units, 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.
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.
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.
Hi Hexagon, I have to go after watching untill 16:54 but I will come back to watch the rest. I want to congratulate you on this beautiful work. I already watched several videos on the Riemann Zeta function but you managed to push it a little further so we lay people can understand it more deeply. Thank You so much and here is my thumb up before I see the remaining part.
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!
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?
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!
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.
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.
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 a fantastic video. I’m not a mathematician, but I’ve been curious about RH for awhile now from a layman’s perspective, mainly a result of reading a book called The Humans and then going down the RH rabbit hole. I’ve read several articles and watched several videos, and I think I had a reasonably good layman’s grasp of the Riemann prime counting function, but one thing I couldn’t get a handle on was just why a real part ½ was so important (as opposed to just any non-trivial zero). This video, with its graphics, does a really good job of simplifying things for people like me - it really does prove the statement that a picture is worth a thousand words. One thing - the video focuses on what happens if zeros are found with real part > ½, but doesn’t really address what happens if < ½. I think it would help to explain that, as John Chessant pointed out in his comment, and to use his wording, “zeros in the critical strip have reflection symmetry across the critical line. eg. 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 ½ that would mess up that x^(½) error bound”. And to expand on that a bit more (hopefully someone will correct me if I’m wrong about this), at 12:04 in the video, waves with the same imaginary part but different real parts have the same frequency but different amplitudes. So not only would the wave with the real part larger than ½ mess with the bound, I think when you add the smaller wave too (as described at 12:37), that would mess it up even more. (Again, I hope I’ve got that right.)
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
Amazing explanation, also great flow of ideas through the video. Also, nicely use of graphics ! Thanks for sharing this amazing knowledge with non-mathematicians! 🔥
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.
calling back to Fermat and his last theoram... eh? As ya know, it took nearly 300 years after finally Andrew Wiles proved it in 1994 and it took him 129 pages... I can only guess how long your proof is, I'm afraid that you will fail to write it down beside the margine of a math text book 😂
@@louisrobitaille5810 Don't ye lose hope. There are 6 dang millennium problems left... The horse is not dead yet, plenty of beatings left until it's demise.
Dear noble friends of this simple page, I apologize if the numbers I mentioned are not prime, and the exact and non-exact roots are equal to the enigmatic number of pi that I standardized (3.15), thus this "Hypothesis Riemann completely loses its strength in the theories of past times, and in the current era this enigmatic number of pi has been standardized to be Rational and Irreversible with a fraction of whole numbers, with two beautiful standardized formulas....
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)?
"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.
The integral representation shown here is actually derived using the infinite series formula for the zeta function, so it is also only defined for Re(s) > 1.
One of the integral representations that is actually defined for all s =/= 0 uses the Abel-Plana formula and you can type it into Desmos if anyone wants to play around with the function.
Something is confusing at 5:00, since the definitions of ζ(s) and Γ(s) are only given by the displayed integrals for Re(s) > 0, and otherwise you need analytic continuations.
Dear Hex ~ You may agree that Mathologer rocks the "world" of maths to the core, re: the video on Pythagorean geometry & Fibonacci numbers (etc.). For example, I now see the way to use geometry + graphic programmiong to find the exact location of each primal positive number ('prime') n (=p) in the sequence n + 1 of N => positive infinity. For example, since all p = 6n +/-1 and there are only primals, coprimes, and pseudo-primal composites at 6n +/-1 then, in any decan of N at magnitude/cardinality M/C, we can check for primality by using the pythagorean-fibonacci geometry (PTG) rule. In other words, by progressing along the number line of N+ (or R+), we can eliminate multiples of n & p, yet also check for primality at 6n +/-1 by using the PTG Rule. Voila! We find no mystery of primal numeric logic or locations of noncomopsites p, and no mysterious patterns of p (determined by the symmetries and regularities of the preceding composites n). Clearly, this verifies my 2017 insight (& mapping). The noncomposites p are gaps in the sequences of composites n, due to the result of dyadic arithmetic continuation of n + 1. This also confirms the intrinsic interdependence of geometry and "numbers" as expressions of geometric-numeric logic, enabled by the natural metalogical principles of being (the cosmos, or life). QED. For more extensive consideration andf/or discussion, see my preprints (at ResearchGate .net). Thanks & best of luck etc. ~ M
He did talk about that? Said we are far from it, although many believe it to be true because of the trillion zeros. However, a good reason even that is not a good reason is the Borwein Integral. In which it looks convincing that the integral stays at π/2 but at a large number it suddenly stops doing that.
I think you need to think a bit more about what your audience is; to understand this video at all it is necessary for someone to have taken an algebra class (note the frequent use of the terms "solutions", "functions", etc)- I don't think it's unreasonable to expect the viewer to already understand imaginary numbers, especially if all of the other terms aren't explained. I enjoyed your cheeky pi^2/6 at 16:33!
I disagree - laypeople interested in math have an intuitive understanding of ubiquitous terms like "solution" and (to a lesser extent) "function" (although "function" would be equally well understood by a german for example :p), whereas if someone doesn't know what an imaginary number is then they simply do not know because it is an abstraction which one needs to be taught, rather than an intuitive notion that can be picked up.
@@schweinmachtbree1013 I mean are you sure it's like someone told you you got two apples you be like okay and then some other time they told you you got negative two apples now you know basic math to minus two two plus two and you got no apples - 2 well imagine that it seems like I can't do that not in the real world so you got to do it inside your head and that's how maybe someone that perhaps might not even know the word for imaginary numbers you just know like a concept and someday we go to school go to college learning notation and now they can put down something in a language meant to last thousands of years because I mean we can go back and open up principia Mathematica or you know something about Euler you know something maybe you could wrote it's not saying you could Euclid there you go sorry I'm doing voice to text because it's quiet in here and I had kind of a bad afternoon so it's easier to talk out loud say a nice things to other people try to encourage people and then since I'm saying it thinking it hearing it you know it's reinforcing this inside my own head so this is good therapy for me and also it's happening where I'm conceptually thinking about his ideas so it's possibly I don't know but if you follow the idea of tangential thinking, perhaps more activity in that area I don't know maybe I'll get a little bit better in math and I'll be able to understand more of those terms as I go along and as I learn.
Perhaps you already knew this and put it in as a joke, but the equation you put in at the beginning (x^5 - x - 1 = 0) is “solvable” so long as you use Bring radicals. In fact all polynomials are “solvable” so long as you invent new functions in exactly the same way that roots were invented to solve polynomials of degree 2-4.
You need to be precise about mathematical statements. To wit, polynomials with only real coefficients are being discussed. If that is the case then polynomials of degree 5 or more are not "solvable", i.e one cannot write down a formula with addition, multiplication or radicals like you can for the quadratic, the cubic, or the quartic. However, this does not preclude formulas for specific polynomials or a whole class of polynomials.
i'll tell you something creepy... the elementary function, Sin(z) can be written in terms of, reflections of Zeta, and reflections of the Gamma function. (this can be derived somewhat easily from the Zeta functional equation). All trig/hyperbolic functions are just rotations of one specific manifold.
In ancient times, Eratosthenes' sieve did shine, A mathematical marvel, a treasure divine. With keen insight, he sieved primes apart, Unveiling patterns with his intellectual art. Through the ages, his legacy held tight, Guiding minds to realms of mathematical light. In numbers vast, his sieve did gleam, Revealing primes like a radiant dream. And in the depths of Riemann's mind did stir, A converter of primes, a theorem so pure. With zeta's function, he charted the way, To understand primes in a mystical array. Through complex contours and analytic finesse, He probed the depths of the number's caress. In Riemann's realm, the primes did unfold, In a dance of zeros, a story untold. So let us honor these mathematical sages, Whose brilliance echoes through the ages. In Eratosthenes' sieve and Riemann's converter, Lies the beauty of numbers, forever and ever.
It does. Although I wouldn't write x+0*i for a real number, because that would still assume that its zero-element can be multiplied with i, whereas that operation isn't defined within the reals. But I get what you mean. The real numbers are homeomorphic to the complex numbers with imaginary component zero, or some lingo like that.
This is not about solving an equation, which is obviously determinable by plugging in the appropriate numbers, but proving an assertion. Different concepts.
Amazing video, but just one comment or maybe a possible answer of yours. As long as I know, the way you are defining the zeta function with that integral being divided by the gamma function, also only converges if R(s)>1. It would be nice if you add that in the description of the video. Best regards.
Riemann Converter? Well, I have read four books on the Riemann Hypothesis: 1)-Prime Numbers and the RH by Barry Mazur and William Stein; 2)-Music of the Primes by Marcus du Sautoy; 3)-Stalking the RH by Dan Rockmore; 4)- Prime Obsession by John Derbyshire. None of these authors mentions anything about a "Riemann Converter". Nevertheless, this presentation takes the layman who doesn't want to worry too much about details to the heart of the Riemann Hypothesis in just under 29 minutes. This is no mean feat!
Seems to me one (easy) way to approach the proof is to show there is no approximation to the prime counting function that is as good as what Riemann Hypothesis would imply.
primes are so interesting to me because they are not just fundamental to the universe but to reality itself. even if the universe restarted with completely different laws of physics, as long as you have discrete objects to count and space to arrange them in rectangles, primes are there. but we know so little about them, why should they exist?
Nice video. So there is unbounded growth in the oscillation if there is a zero with Re(s) not 1/2 but < 1. But wouldn't oscillations cancel with symmetric zeros about Re = 1/2? Say Re(s) = 0.7, Re(s) = 0.3
im no mathematician but this seems like inherrent logic to me. 1) Primes "encode" how numbers work. there have to be fiewer as you move on because there are more divisors. the exact rules give us the prime distribution. it literally cant be diffrent or the rules must be diffrent. 2) so this means the form and rythem (amplitude and frequency) are directly dependent on the literal rules of mathematics and cant change. because if the primes where any diffrent they would be divisible by 2 (for example) - so slope and freq will never change. thats like literally a prove that the function cant be different from that 0,5 line at pricise steps. be left or right of the line or not in Sync with the "up steps" (i direction) would mean that the function look diffrent and mathematics of primes would be screwed.
Proving the Riemann hypothesis: Moments after a Big Bang, dual numbers are at the origin, repulsive and gravitational forces are balanced, but dark matter is breaking up. Delanges sectrices for the unflaking, Archimedean spiral for spinning up dark matter, Ceva sectrices for that dark matter breaking into smaller, primordial dark matter and black holes, Dinostratus quadratrix for impacting surrounding black holes, and Maclaurin sectrices for dark matter being slowed down over time. Split-complex at one, noting the more gravitational nature of diagonal ring/cylinder singularities of baryonic matter from broken dark matter. Using repetitious bisection, one gets added (SSA) for when dark matter dominates a universe and one gets added for when black holes and rarified singularities make things more gravitational, leading to a Big Crunch. At 2 = a Big Crunch, the solution to the Basel problem, and the "bellows method" drives things. A Big Crunch event needs the net gravity from the localized area AND gravitational effects from surrounding universes. The 2 represents how locked up black holes are making a stacking, planar contact. Everything else are harmonics - effects on surrounding universes. Ramanujan's Infinite Sum of negative one-twelfth: each positional jostling, mass, and gravitational/attractive exchange between universes (defined by Big Bounce events, considering Ramanujan during both Big Crunch and Big Bang sides) ADDS up to evenness (or a partner in marriage - like half of the whole, I guess). Net gravity increases over time from the breaking up of dark matter, much like a vertical asymptote. It continues to increase, but hits a critical point, much like a vertical tangent, when enough dark matter has weakened and been broken up into the more gravitational baryonic matter (split-complex numbers). Then, the effects of gravity reach their upper limit - much like a vertical line - at e to the negative e. Over a decade ago, I said that the potential energy of a Big Bang was 1/6 times (the speed of light, cubed). If I had said Big Bounce instead, I would have been correct.
It is kind of intriguing that offering a financial reward for a solution that satisfies hypothetical Conjectures is to teach the "price of everything and the value of nothing", by default, ..a kind of "Precision is not Accuracy", redefinition.
I admit something I have been curious about with the reimann hypothesis is if the non trivial zeros are potentially simply a consequence of the fact that complex numbers indirectly produce negatives . All the trivial zeros after all are negatives that will be switch around to be a non negative variation in n instead of 1/n Beyond that perhaps ities into the squaring uniqueness of 1/2 versus other numbers
Welcome to 'secrets of the universe'. after solving the Riemann Hypothesis we will also learn how to fabricate plutonium for our flux capacitors, using regular household items.
“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!
lmaooooo
If I was introduced complex numbers like this, I would quit math
😃😃😃
School detected
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
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
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
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.
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.
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.
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!!!
This is the best video on Riemann hypothesis I've seen on YT. Congratulations on explaining it in-depth yet in simple terms.
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!
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!
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've accidentally proven that li(x) and pi(x) are inseparable. I don't know what to do about it, I don't want to be called a crank or a crackpot, it was an accident. If there's an expert in this problem that is interested, I can talk you through my process, maybe you could confirm this for me?
It would be easier than multiple rejections, I don't have credentials, instead I have extreme imposter syndrome and a solution to this problem. I don't know what to do. I really could use some help and advice.
Also, it's extremely simple, but it is time consuming.
I've accidentally proven that li(x) and pi(x) are inseparable. I don't know what to do about it, I don't want to be called a crank or a crackpot, it was an accident. If there's an expert in this problem that is interested, I can talk you through my process, maybe you could confirm this for me?
It would be easier than multiple rejections, I don't have credentials, instead I have extreme imposter syndrome and a solution to this problem. I don't know what to do. I really could use some help and advice.
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.
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.
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 🤦🏼♂️
Fantastic video! I have watched multiple TH-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!
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 1 and _p_ , up to units,
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.
This was very well presented, and honestly, I didn't think anybody would do it better than 3b1b, but you done did it...
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.
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.
Hi Hexagon, I have to go after watching untill 16:54 but I will come back to watch the rest. I want to congratulate you on this beautiful work. I already watched several videos on the Riemann Zeta function but you managed to push it a little further so we lay people can understand it more deeply. Thank You so much and here is my thumb up before I see the remaining part.
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.
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!
Totally watchable, well done in showing how to present clearly a complex subject to anyone, thank you.
The most accessible explanation I've ever seen (from someone that has a bit of maths). Congratulations and 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?
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!
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!
This is the best Maths video I have seen on TH-cam. Well done.
Thank you, this helped me 'understand' the Riemann Hypothesis much better than anything else I've encountered.
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.
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.
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?
Best and most accessible summary of the subject I have seen. Great graphics!
This is amazing! Hands down best video on the topic I've seen (and that means better than 3b1b which is saying something!)
Math video of the year. Finally someone who explains the big deal!
This is a fantastic video. I’m not a mathematician, but I’ve been curious about RH for awhile now from a layman’s perspective, mainly a result of reading a book called The Humans and then going down the RH rabbit hole. I’ve read several articles and watched several videos, and I think I had a reasonably good layman’s grasp of the Riemann prime counting function, but one thing I couldn’t get a handle on was just why a real part ½ was so important (as opposed to just any non-trivial zero). This video, with its graphics, does a really good job of simplifying things for people like me - it really does prove the statement that a picture is worth a thousand words.
One thing - the video focuses on what happens if zeros are found with real part > ½, but doesn’t really address what happens if < ½. I think it would help to explain that, as John Chessant pointed out in his comment, and to use his wording, “zeros in the critical strip have reflection symmetry across the critical line. eg. 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 ½ that would mess up that x^(½) error bound”. And to expand on that a bit more (hopefully someone will correct me if I’m wrong about this), at 12:04 in the video, waves with the same imaginary part but different real parts have the same frequency but different amplitudes. So not only would the wave with the real part larger than ½ mess with the bound, I think when you add the smaller wave too (as described at 12:37), that would mess it up even more. (Again, I hope I’ve got that right.)
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.
very well explained - first time i've understood any part of the Riemann Hypothesis!
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
Amazing explanation, also great flow of ideas through the video. Also, nicely use of graphics ! Thanks for sharing this amazing knowledge with non-mathematicians! 🔥
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.
Incredible video. I now understand the importance of RH so much better. Thanks man
Best video on this topic so far... thank you!
i have proof. but theres no space in the comments to show you.
😆
calling back to Fermat and his last theoram... eh?
As ya know, it took nearly 300 years after finally Andrew Wiles proved it in 1994 and it took him 129 pages...
I can only guess how long your proof is, I'm afraid that you will fail to write it down beside the margine of a math text book 😂
@@munimahmed7877 It would've been hilarious if Fermat had been wrong and people struggled forever 😂.
@@louisrobitaille5810 Don't ye lose hope. There are 6 dang millennium problems left... The horse is not dead yet, plenty of beatings left until it's demise.
Oh that's mean! Lol
I love this explainer because it's the only one I've seen talking about the Fourier series aspect. Thank you!
Dear noble friends of this simple page, I apologize if the numbers I mentioned are not prime, and the exact and non-exact roots are equal to the enigmatic number of pi that I standardized (3.15), thus this "Hypothesis Riemann completely loses its strength in the theories of past times, and in the current era this enigmatic number of pi has been standardized to be Rational and Irreversible with a fraction of whole numbers, with two beautiful standardized formulas....
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)).
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)?
"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.
The integral representation shown here is actually derived using the infinite series formula for the zeta function, so it is also only defined for Re(s) > 1.
One of the integral representations that is actually defined for all s =/= 0 uses the Abel-Plana formula and you can type it into Desmos if anyone wants to play around with the function.
These are some of the crispest animations I have seen in my life, bravo.
Something is confusing at 5:00, since the definitions of ζ(s) and Γ(s) are only given by the displayed integrals for Re(s) > 0, and otherwise you need analytic continuations.
i find it incredibly insulting that this video only has 30k views
By far the best video on the RH on youtube! Thank you :)
I worked out the patton in 1978 and now have it linked to pi/sinewaves and the link between two primes.
pls make a longer video going into the things you couldn't
Dear Hex ~ You may agree that Mathologer rocks the "world" of maths to the core, re: the video on Pythagorean geometry & Fibonacci numbers (etc.). For example, I now see the way to use geometry + graphic programmiong to find the exact location of each primal positive number ('prime') n (=p) in the sequence n + 1 of N => positive infinity. For example, since all p = 6n +/-1 and there are only primals, coprimes, and pseudo-primal composites at 6n +/-1 then, in any decan of N at magnitude/cardinality M/C, we can check for primality by using the pythagorean-fibonacci geometry (PTG) rule. In other words, by progressing along the number line of N+ (or R+), we can eliminate multiples of n & p, yet also check for primality at 6n +/-1 by using the PTG Rule. Voila! We find no mystery of primal numeric logic or locations of noncomopsites p, and no mysterious patterns of p (determined by the symmetries and regularities of the preceding composites n). Clearly, this verifies my 2017 insight (& mapping). The noncomposites p are gaps in the sequences of composites n, due to the result of dyadic arithmetic continuation of n + 1. This also confirms the intrinsic interdependence of geometry and "numbers" as expressions of geometric-numeric logic, enabled by the natural metalogical principles of being (the cosmos, or life). QED. For more extensive consideration andf/or discussion, see my preprints (at ResearchGate .net). Thanks & best of luck etc. ~ M
I would have liked to see a discussion of how close RH is to being true: an infinite number of zeros, at least a finite fraction of the zeros, etc.
He did talk about that? Said we are far from it, although many believe it to be true because of the trillion zeros. However, a good reason even that is not a good reason is the Borwein Integral. In which it looks convincing that the integral stays at π/2 but at a large number it suddenly stops doing that.
I think you need to think a bit more about what your audience is; to understand this video at all it is necessary for someone to have taken an algebra class (note the frequent use of the terms "solutions", "functions", etc)- I don't think it's unreasonable to expect the viewer to already understand imaginary numbers, especially if all of the other terms aren't explained.
I enjoyed your cheeky pi^2/6 at 16:33!
I disagree - laypeople interested in math have an intuitive understanding of ubiquitous terms like "solution" and (to a lesser extent) "function" (although "function" would be equally well understood by a german for example :p), whereas if someone doesn't know what an imaginary number is then they simply do not know because it is an abstraction which one needs to be taught, rather than an intuitive notion that can be picked up.
@@schweinmachtbree1013 I mean are you sure it's like someone told you you got two apples you be like okay and then some other time they told you you got negative two apples now you know basic math to minus two two plus two and you got no apples - 2 well imagine that it seems like I can't do that not in the real world so you got to do it inside your head and that's how maybe someone that perhaps might not even know the word for imaginary numbers you just know like a concept and someday we go to school go to college learning notation and now they can put down something in a language meant to last thousands of years because I mean we can go back and open up principia Mathematica or you know something about Euler you know something maybe you could wrote it's not saying you could Euclid there you go sorry I'm doing voice to text because it's quiet in here and I had kind of a bad afternoon so it's easier to talk out loud say a nice things to other people try to encourage people and then since I'm saying it thinking it hearing it you know it's reinforcing this inside my own head so this is good therapy for me and also it's happening where I'm conceptually thinking about his ideas so it's possibly I don't know but if you follow the idea of tangential thinking, perhaps more activity in that area I don't know maybe I'll get a little bit better in math and I'll be able to understand more of those terms as I go along and as I learn.
I misread the title as "The Riemann Hypothesis failed" and along with the thumbnail I thought they'd found a counter-example 😭😭💔
same lol.
I really like the way you presented this. Mentioning Sheldon twice is a plus.
I have been looking for something like this for so long
Thanks! 🙏
Perhaps you already knew this and put it in as a joke, but the equation you put in at the beginning (x^5 - x - 1 = 0) is “solvable” so long as you use Bring radicals. In fact all polynomials are “solvable” so long as you invent new functions in exactly the same way that roots were invented to solve polynomials of degree 2-4.
Perhaps not. This is not very widely know, and not covered in even college courses. At least not that I've seen.
You need to be precise about mathematical statements. To wit, polynomials with only real coefficients are being discussed. If that is the case then polynomials of degree 5 or more are not "solvable", i.e one cannot write down a formula with addition, multiplication or radicals like you can for the quadratic, the cubic, or the quartic. However, this does not preclude formulas for specific polynomials or a whole class of polynomials.
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 this video. It really helped me understand more about this problem - although still a lot I don't yet fully get!
Great video! I finally get how the zero's of the Zeta function relate to the prime numbers.
i'll tell you something creepy... the elementary function, Sin(z) can be written in terms of, reflections of Zeta, and reflections of the Gamma function. (this can be derived somewhat easily from the Zeta functional equation). All trig/hyperbolic functions are just rotations of one specific manifold.
In ancient times, Eratosthenes' sieve did shine,
A mathematical marvel, a treasure divine.
With keen insight, he sieved primes apart,
Unveiling patterns with his intellectual art.
Through the ages, his legacy held tight,
Guiding minds to realms of mathematical light.
In numbers vast, his sieve did gleam,
Revealing primes like a radiant dream.
And in the depths of Riemann's mind did stir,
A converter of primes, a theorem so pure.
With zeta's function, he charted the way,
To understand primes in a mystical array.
Through complex contours and analytic finesse,
He probed the depths of the number's caress.
In Riemann's realm, the primes did unfold,
In a dance of zeros, a story untold.
So let us honor these mathematical sages,
Whose brilliance echoes through the ages.
In Eratosthenes' sieve and Riemann's converter,
Lies the beauty of numbers, forever and ever.
At 18:46 , It should be noted that the 1 million dollar prize would not be awarded to someone disproving the conjecture.
Heartbreaking, but also how I read the prize rules.
I thought it went like this:
Real numbers: x+0*i
Imaginary numbers: 0+x*i
Complex numbers: x+y*i
It does.
Although I wouldn't write x+0*i for a real number, because that would still assume that its zero-element can be multiplied with i, whereas that operation isn't defined within the reals.
But I get what you mean. The real numbers are homeomorphic to the complex numbers with imaginary component zero, or some lingo like that.
make a vid that expands and goes deeper into everything you said here
This is not about solving an equation, which is obviously determinable by plugging in the appropriate numbers, but proving an assertion. Different concepts.
Amazing video, but just one comment or maybe a possible answer of yours.
As long as I know, the way you are defining the zeta function with that integral being divided by the gamma function, also only converges if R(s)>1. It would be nice if you add that in the description of the video.
Best regards.
This is the first video to make me understand how everything is really connected.
This is the first video where all of the comments praising your easy to understand explanation where actually correct
Thank you!
I've never seen this explained so well.
Man oh man...brilliant presentation.....please keep up!!
14:29 I do know that this math and science. But this harmonics are f****g black magic. Math is amazing
Riemann Converter? Well, I have read four books on the Riemann Hypothesis: 1)-Prime Numbers and the RH by Barry Mazur and William Stein; 2)-Music of the Primes by Marcus du Sautoy; 3)-Stalking the RH by Dan Rockmore; 4)- Prime Obsession by John Derbyshire. None of these authors mentions anything about a "Riemann Converter". Nevertheless, this presentation takes the layman who doesn't want to worry too much about details to the heart of the Riemann Hypothesis in just under 29 minutes. This is no mean feat!
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
This is the best video on this topic !!!
Seems to me one (easy) way to approach the proof is to show there is no approximation to the prime counting function that is as good as what Riemann Hypothesis would imply.
could you provide the source for the Riemann converter function?
wow, so 0.5 +10i is a kickdrum. Neat!
yup, my takeaway is that the Riemann converter basically is a drum machine
Best Zeta explainer yet. 👍
primes are so interesting to me because they are not just fundamental to the universe but to reality itself. even if the universe restarted with completely different laws of physics, as long as you have discrete objects to count and space to arrange them in rectangles, primes are there. but we know so little about them, why should they exist?
This was brilliant! Thank your for the great video
Nice video. So there is unbounded growth in the oscillation if there is a zero with Re(s) not 1/2 but < 1. But wouldn't oscillations cancel with symmetric zeros about Re = 1/2? Say Re(s) = 0.7, Re(s) = 0.3
cannot understand a single idea but still watching
im no mathematician but this seems like inherrent logic to me.
1) Primes "encode" how numbers work. there have to be fiewer as you move on because there are more divisors. the exact rules give us the prime distribution. it literally cant be diffrent or the rules must be diffrent.
2) so this means the form and rythem (amplitude and frequency) are directly dependent on the literal rules of mathematics and cant change.
because if the primes where any diffrent they would be divisible by 2 (for example) - so slope and freq will never change.
thats like literally a prove that the function cant be different from that 0,5 line at pricise steps.
be left or right of the line or not in Sync with the "up steps" (i direction) would mean that the function look diffrent and mathematics of primes would be screwed.
Congratulation! excellent explanation
Proving the Riemann hypothesis: Moments after a Big Bang, dual numbers are at the origin, repulsive and gravitational forces are balanced, but dark matter is breaking up. Delanges sectrices for the unflaking, Archimedean spiral for spinning up dark matter, Ceva sectrices for that dark matter breaking into smaller, primordial dark matter and black holes, Dinostratus quadratrix for impacting surrounding black holes, and Maclaurin sectrices for dark matter being slowed down over time. Split-complex at one, noting the more gravitational nature of diagonal ring/cylinder singularities of baryonic matter from broken dark matter. Using repetitious bisection, one gets added (SSA) for when dark matter dominates a universe and one gets added for when black holes and rarified singularities make things more gravitational, leading to a Big Crunch. At 2 = a Big Crunch, the solution to the Basel problem, and the "bellows method" drives things. A Big Crunch event needs the net gravity from the localized area AND gravitational effects from surrounding universes. The 2 represents how locked up black holes are making a stacking, planar contact. Everything else are harmonics - effects on surrounding universes.
Ramanujan's Infinite Sum of negative one-twelfth: each positional jostling, mass, and gravitational/attractive exchange between universes (defined by Big Bounce events, considering Ramanujan during both Big Crunch and Big Bang sides) ADDS up to evenness (or a partner in marriage - like half of the whole, I guess).
Net gravity increases over time from the breaking up of dark matter, much like a vertical asymptote. It continues to increase, but hits a critical point, much like a vertical tangent, when enough dark matter has weakened and been broken up into the more gravitational baryonic matter (split-complex numbers). Then, the effects of gravity reach their upper limit - much like a vertical line - at e to the negative e.
Over a decade ago, I said that the potential energy of a Big Bang was 1/6 times (the speed of light, cubed). If I had said Big Bounce instead, I would have been correct.
It is kind of intriguing that offering a financial reward for a solution that satisfies hypothetical Conjectures is to teach the "price of everything and the value of nothing", by default, ..a kind of "Precision is not Accuracy", redefinition.
I admit something I have been curious about with the reimann hypothesis is if the non trivial zeros are potentially simply a consequence of the fact that complex numbers indirectly produce negatives . All the trivial zeros after all are negatives that will be switch around to be a non negative variation in n instead of 1/n Beyond that perhaps ities into the squaring uniqueness of 1/2 versus other numbers
Thanks for the video. It is both accessible and in-depth.
Very well explained and thrill conveyed :)
Welcome to 'secrets of the universe'. after solving the Riemann Hypothesis we will also learn how to fabricate plutonium for our flux capacitors, using regular household items.