At 23:26 it says “This video is part of PeakMath course. Join the journey at …” same thing in video description. So I wouldn’t hold my breath because PeakMath is a subscription service. This is probably an advertisement of sorts.
@@InShadowsLinger Not quite. All these Episodes will be freely available on RU-vid, and we have planned at least five full Seasons, each with 10 to 12 Episodes. But there will also be an online course community running in the background for those who find value in reading extended written course notes and having direct access to the two of us for math questions. We're only just getting started this week, and haven't figured out all the details yet. But this math content is the stuff of dreams, and I hope we can offer something of value in the course that goes beyond what you could possibly do by pure YouTubing.
I am a complete layman and in fact a high school dropout. I try to catch up to maths and other subjects in my spare time, and I don't shy from esoteric or advanced topics such as the Reimann Hypothesis because of the simple fact that all the other people who discovered these things were also humans. Your way of explaining is one of the most easy to follow. A nice mirror to the other awesome RU-vidrs like 3Blue1Brown. Whereas they try to beautifully simplify things from alternative points of view, you approach the problem head on with clarity and patience. Thanks!
Awesome video! I'm very happy RU-vid recommended this to me. I always wanted to understand how the Riemann Zeta function is related to prime numbers and you explained it so clearly and easy to understand. Keep up the great work!! I'm really looking forward for the next episodes of this series :D
Best math video on RU-vid this year. I've always marvelled at how productive Archimedes was with such a limited notion of number that I decided maybe my notion of number was the problem. We treat transcendentals like integers all day long and never bat an eyelid. Have you ever done a calculation with pi? No you haven't.
This is very high quality content. I am a first year mathematics undergraduate and appreciate that I can understand concepts of such advanced level because of the way you dissect each and every part. Please continue these videos!
The examples at 8:49 made it very clear to me exactly the algebraic numbers one finds in P, such a great video series. Rewatching now that I know more than I did before, and will continue rewatching until I understand everything in this video. I want to prove myself right. Thank you for your content, sir.
I'm a mathematics graduate student right now working in non-commutative geometry, operator theory, and field theories. I'm not particularly interested in working on the Riemann hypothesis myself, but (while not necessarily frequent) I do recurringly hear and read about how the the stuff I work with has various sorts of connections with the Riemann hypothesis, and I do find that rather fascinating I hope this series continues! Though I don't plan on ever working on the Riemann hypothesis myself, I'm excited at the prospect of hearing an extended pressntation which elabourates on some of those connections I've heard about!
@@extreme4180 That's probably not the best wording ever for asking a question to a stranger... But if you are worried about income during grad school, in my case I lived off scholarships and part-time jobs when I did my PhD (I could barely pay my power bills, well I was actually unable to do so a couple times). Many other grad students do the same.
@@pseudolullus um sorry, so as to pursue maths i need other source of incomes i guess, i'm still in highschool so i need to make a plan or get a good university within my country ... thnx
@@pseudolullus as an indian resident, our govt. is constantly supporting the field of research so i can stand a chance , is there any way to connect with you online?
Physicist chipping in here. You are putting the bar high up there, love it! I look forward to all the next episodes. Hope you will extensively cover random matrix theory as well.
Sidebar: I'm too bogged down doing yard work to think about any of this stuff using a well rationed bottom up incremental procedure, but, when I see a list I always want to sort it vertically within 25 orders of complexity to see if I can convince myself of soundness and its' derivatives, and laterally with circular relationality to convince myself there is tension and proximity to a singularity. After that, all proofs should be local to the spread, unless captured by new top down notation or circumstance. It's like lying about everything, untill one finds the missing link that connects everything to the core spectral tree. To maintain interest, I might try to plot the args between the non trivial and discovered half zeros, just to get that notion out of the way. After that, I might try to see if prime spirals pass through zeros. Beyond that, I might try a complex basis, for a complex function, instead of just relying on the real natural. After 6 months, and all that snow shovelling and lawn mowing, if I hadn't got anywhere, I'd give it a rest, and move to plan B? In plan B, I might generalize complex basis into radial or steridian arms, and choose the one that stayed true the longest, or offered the possibility of predictively jumping basis, to a new complex basis? Plan C, now 2 years down the road, and retail computing aside, pocketing positional bread crumbs just might produce a computational thread, or confinement proof, which brackets expectation, and a re-sort of prior lists? Plan D, sluffing off real mathematicians who incrementally produce the notation for real thoughts, rather then guys that just count to the limits of thought and contradiction. Plan E, transforming zeta into a torus and coloring the holes, and then trying aspect projections to line them up into some spirals? Oh well, just a thought that paper hanging isn't going to work, but optical discovery might. Plan F, predicting a change of basis to find the next conformity to consistency, and maybe discovering a hidden singularity that requires discovery of its' charme, before categoricalicity can be claimed. Plan G, compactness still illusive, but completeness expected any day now, after 6 years, ...dwelling now on shadow biological spectrums that don't actually exist other then in the mind, but which can be calculated in digital form, and thinking these are the numbers that are actually also invisible functions, and singularities that have always been taken for granted? Plan H, staying away from psychiatrists who insist everything is real, otherwise you're living in a delusion, and soon to be placed under administration, just so the government can claim patent discoveries they may never have any possible valid natural title to, other then from escheats by divorce, estate, and probate lawyers who are also impunity criminals for publically elected celebrities? Plan I, retracing the path of "i" before it became a singularity operator? It's year 7 now, ...and the city is entertaining complaints that a hermit hadn't been mowing his lawn, and the yard is an eyesore, and there should be a law against people like that. Plan J, discovering that sequenced 2 way spirals, vertical and horizontal, are way better at prediction, just perfectly matched to new optical retail computers that are a billion times faster then they were 10 years ago, and can 3D print lenses for formating 3D schartz christoffel constructors that prove a whole new branch of discovery unknown, but predicted say 8 years ago? Plan K, the realization that the general construction company had a singing frog that only performed on Sundays, every other prime week during some, but not all moon phases where saturn and jupiter were in each other's conflict. Oh well, the city is now threatening to claim the property for lack of tax payments, but completeness is still an unknown incompleteness witch, and a common half bitch, and those orders of complexity to complete, now exceed the mental capacity of every researcher to date, including the latest persistent believer. Plan L, figures that incompleteness on this proof might exceed the natural spectrum, and that augmented cosmic metals are required from yet to arrive space junk to justify and prove realization of the previously invisible. Plan M, praying to the void and expecting a delivery of new resources from a worm hole. Plan N, joining a registered arctic coven that licks frogs for inspiration. Plan O, kneeling next to a tall tree and passing in the zen forest. Well, that's it. Good luck.
Very interesting, please continue the series. 😊 A short personal anecdote, because I thought you were going here when connecting the primes to the Riemann zeta function: Some 38 years ago, with only freshman physics education, and never having heard of Basel problem or Riemann ζ, I was fiddling about with factorisations and found that the product Π 1/(1-p^(-k)) over all primes had to be equal to the sum Σ 1/n^k over all integers (for some fixed k>1). My dad saw my notes and wanted to show it to a mathematician friend of his. I protested a bit, because It was only a few scribbles showing the idea to myself, but he persisted. A few weeks later, I got it back full of red stripes, because my 'proof' was wrong. My dad then lost faith and thought that all of my scribbles were nonsense. Later I found out that Euler had found the same result. Probably the mathematician had thought that I wanted to show a proof for Euler's result, or even trying something with ζ(k), whereas I, naive as I was, just thought to have found an interesting new connection between primes and powers of natural numbers, and had not focused on (and probably could not provide) a watertight proof.
Hi Koen, nice story! Doing a night shift here with the script for Episode 10. I don't know you, but if you are the Koen I believe you are, you will feature in that video, together with Connes-Consani and a few others...
@@PeakMathLandscape Hi, thanks. I don't think that's me. Btw, I am not sure I remember the relation correctly, it may still be wrong. Keep up the good work!
Strong start! I hope future videos, once we've gotten through the big-picture overview, dive deeper into the details and rigor of the problem. In the meantime, I appreciate the links to the papers youve cited!
Nice work! This was the first explanation I've seen of how Riemann actually relates to the primes, via the Riemann spectrum and the cos(K ln(x)) sums... I love when I see new material like this when I know it's been out there all this time. I'll be waiting for your next episodes!
I'm a math's student. and you are the teacher like god. I salute you for introducing this pure Knowlege us. I wish, someday I will go deep down this L-function world and make out something. thank you .
An excellent introduction and overview of the topic. Thank-you so much for taking the time to make such an accessible, well-paced, clear video on the search for proofs of the RH. My hope is that this stimulates young & old mathematicians to embark on this journey of discovery.
Here's a comment on Random Matrices. The coincidences between the asymptotic statistics of eigenvalues of random matrices and zeros of the zeta function are striking. But they are only probabilistic in nature, referring to averages over large sets of random matrices, not to any particular linear operator. At first sight, this only implies that these may somehow belong to the same "universality class", when properly scaled. The use of a spectral approach to proving the RH would have required showing the existence of a specific linear operator, whose discrete spectrum coincides with the zeta function zeros, and which is invariant under an involution whose fixed points are the critical line. Nothing like this has ever been done (for the genuine zeta function) - although analogs for prime number fields have been used to prove the RH in those cases. So the resemblance to the statistics of the eigenvalues of large random matrices is perhaps helpful in suggesting such asymptotic properties for the zeta function zeros, but it does not mean that a spectral approach provides the key to proving the RH. However, it is an appealing idea that, especially for physicists familiar with scattering theory, is very natural, and perhaps should be further pursued. It is just that the class of linear operators used in proving the RH over mod p finite fields, which were discrete analogs of the Laplacian, need not be the same as the one required to prove it for the usual zeta function.
Hi John, thanks for commenting here! The next few episodes will still be very introductory, but after that there might be a few ideas of interest to you. At least I hope so. One of the many "clues" we will pursue eventually is the idea that F1-geometry perhaps shouldn't be modelled on varieties/schemes (like most of the F1-attempts so far), but more on something like the category of branched coverings of some "arithmetic" analogue of the Riemann sphere. Then operations on L-functions (like tensor product or symmetric powers) would (away from the ramification locus) then simply be fibrewise operations on finite sets, the degree of the L-function would be the generic size of the fibre, etc. In this "arithmetic Riemann sphere" the primes should somehow be included among the points, but I think Spec Z is too rigid to be the right object. Something else is needed, and Spec Z would just not be in the picture at all. Hope to be in touch with you in some form once we get further into these details. And any further comments you may have along the way are always much appreciated.
I'm impressed by the effort to look at numbers and RH at a different perspective, given RH has been around for more than 160 years and yet to be proven!
It's beyond amazing to see the mathematics of reality. It's incredible to see the very building blocks of our existence, far beyond the quantum level - in number form. Wow.
I’ve seen lots of content on the RH, but have never seen someone explain ACTUALLY how the zeros on the critical strip related to the primes! That’s amazing!
I will look forward to the continuation. I liked the measured pace with which the main points were introduced. And I also appreciated the "focussing on the key points" in the Manin and Conrey papers. But, of course, as you said, this was just a guided tour, and not yet an explanation. If you could render Manin's paper more accessible, that would be of great value. Conrey's result doesn't go any way towards a proof of the RH, of course, so that may be interesting, but only of secondary interest. The relation to Random Matrices is also nice, but goes no way towards a proof of the RH. I will follow with a comment on that.
i dont know anything about math but the sum of negative cosines feels almost like a transcendental number like e, getting closer and closer with each continuation of a function, never reaching its full value. its like one step forward from a number. this comment is just for the perspective of what people with only basic math knowledge might think of this video.
This video is a thing of beauty. The presenter's approach was just right and I even loved his long pauses. The tablet was a perfect choice, while the hand drawn notes humanised it. I was drawn in to the subject matter and felt engaged from the outset, as if I was part of the adventure. The only thing I didn't like was that this was the only video on the channel. Please can we have more like this.
Absolutely loved this introductory video! You're a great storyteller and have me really excited for the future episodes! As a math student, I love seeing stuff like this
One of the best introductions I've seen to prime numbers and RH. Thank you! As a lay person I'd like to ask something. To me, there clearly is some sort of geometry to numbers, they are a geometry. One we seem able to just see, as far as we look at numbers and what they do. All it takes is looking. How do numbers stand and behave relative to eachother. Now, one thing is to look at that and describe it, while another is being able to predict things. Why should we be able to predict everything, meaning anything in particular, without computing? We may find a proof of something, some day, or not, but does it imply anything at all about certainty, or rather lack of it, whether we haven't managed yet or it is impossible to prove that? My point is: we might sometimes actually see clearly that which isn't proven or even provable. For instance, the twin primes conjecture. To me it is by definition a conjecture, but at the same time, it seems obviously true, "visually" obvious, to be true, we KNOW it to be true. We can look and see how there will always be twin primes. It is something purely mechanical. We cannot compute all numbers in practice, but we can see how, computing them, necessarily leads to infinite twin primes. It simply is part of their "geometry". My question is: this which I am saying, does it make sense to any mathematician here? Would any of you agree that you can see it (how this conjecture is true) and be just as certain, without a proof? Thank you!
That's so exciting and well presented. Can't wait for the next episode. It might feel a bit like the lecture of Wiles, where he casually proved Fermats last theorem without explicitly telling in the Titel... Greatfully, Alex
This is extremely good quality content. Will you discuss the work of Deninger at some point? Perhaps a discussion of Connes’ noncommutative geometry approach might be worth talking about too!
For ideas and inspiration, yes! But fundamentally we will be looking for geometry that is independent of Spec Z and scheme theory, so not a "compactification" of Spec Z as in many papers of Deninger, and also not a "deeper base" than Spec Z as in many papers of Connes and Consani. In the first case, as I'm sure you know, one imagines an inclusion morphism of Spec Z into the compactification, and in the second one imagines a base change morphism "from F1" to Spec Z. But there could be a geometry living in a completely different category, with no morphisms to or from Spec Z, but where it still makes sense to interpret integers or primes as "functions".
Wow, what a smooth and well behaved explanation. I am quite excited but also alarmed by the 'unfathomable abyss' metaphor. What could possibly be in there at the end???
