I have a gut feeling that you have proved RH is true, and gonna shock the whole world in the last episode 😅
@quarkonium3795
Жыл бұрын
If he's building this series up to explain how he proved the RH then this series would probably be hundreds of episodes long if he's starting out at this basic level
@randomchannel-px6ho
Ай бұрын
On a serious note the tools to solve it's outstanding problems just don't existal and there's zero clues where to possibly develop them and this the topic is of very little theoretical interest because of that. Any indication of that something is possibly a viable tool and it would wuickly matriculate to pop sci
@amritawasthi7030
Жыл бұрын
Top tier content
@NachoSchips
Жыл бұрын
PeakMath Content you might say
@jackr1734
2 ай бұрын
For the average alien
@RSLT
Жыл бұрын
Wow, this video is incredible for anyone interested in the L-functions and Modular Forms Database! It's packed with wonderful resources and information. Thank you for creating and sharing!
@TranSylvainie
Жыл бұрын
Perfection ! Can't wait for the adventure to continue !
@rohitchatterjee2327
10 ай бұрын
I’ve been waiting for a video like this for so long - serious math for a serious audience, but without the formal lemma-proposition-theorem format
@chriscopeman8820
Жыл бұрын
I can follow the steps but the motivation for doing the steps eludes me.
@Craftesteur
Жыл бұрын
Incredible work on incredibly hard stuff!!! Much love from France
@matanfih
Жыл бұрын
Was waiting for it!
@888Xenon
Жыл бұрын
Fantastic stuff, keep it up!
@FreeZh-xyz
Ай бұрын
42:36 Should the generating series be 1 over the product? Thanks
@bini420
Жыл бұрын
Great work. I'll have to start going through the rigorous literature soon, maybe after 2 or 3 videos, to properly understand the things talked in the video and not rely on unstable intuition. Can you link a few articles and books of the like that may prepare me for the topics and fields covered in future videos?
@PeakMathLandscape
Жыл бұрын
There will be carefully selected references in more or less every video. For more specific recommendations, it really depends on what background you already have! One possible starting point for general number theory background (on a more basic level than the L-functions themselves) would be the book "Topology of Numbers" by Hatcher. Many concepts from that book will be used by us, and it's very well written: pi.math.cornell.edu/~hatcher/TN/TNbook.pdf
@bini420
Жыл бұрын
@@PeakMathLandscape thank you, I'll be sure to read through most of the book. Do you have any recommendations for a book on analytic number theory? I went through about the first half of apostols book on it, is there one that aligns with your video more?
@PeakMathLandscape
Жыл бұрын
@@bini420 A standard reference is Iwaniec and Kowalski: Analytic Number Theory. But it is not an easy read. Chapter 5 is about L-functions though, and quite relevant for our videos.
@benpaz9548
Жыл бұрын
we pray for S1E4 🙏
@michaelschnell5633
Жыл бұрын
I answered to my own question regarding the coefficients of the power series approaching for zeta function by providing the URL of a web page that gives them. Unfortunately the KZitem algorithm killed the question and the answer.
@michaelschnell5633
Жыл бұрын
In that article, the Zeta function is given as 1 / ( s-1) + with all coefficients of the power series given by a formula in k. As 1 / (s-1) can be written as a power series in (s) < i.e. sum s**n>), this obviously is a power series in total. I understand that it should be possible to re.base the power series in (s) and in (s-1) to a common base point in C, and hence the coefficients should be known to provide a single globally valid infinite pronominal approximating Zeta. Supposedly this should hold for other analytic L - functions as well.
@Werner-h8m
Жыл бұрын
Wo bleibt die praktische Anwendung zutr Lösung der Gleichungen
@mcmagic6723
Жыл бұрын
At 5:55, you used x^2, in python that means bitwise or, you should've used x**2 for squaring. It's a common mistake in python.
@PeakMathLandscape
Жыл бұрын
Appreciate the comment. But we're using Sage here and not Python, so x^2 is totally fine.
@jarrettreimers2908
Жыл бұрын
Could someone link a video or article that details how to calculate "bad primes"?
@PeakMathLandscape
Жыл бұрын
The idea is to first compute the discriminant of the equation, and then the bad primes are simply the primes that appear among the divisors of the discriminant. A precise explanation of "discriminant" that rigorously covers all the technical details would be very long. But omitting some of these technical details, the discriminant of a degree two polynomial ax^2+bx+c is defined to be b^2-4ac, the discriminant of a general polynomial (still in one variable) is described at Wikipedia: Discriminant, and the discriminant of an elliptic curve is defined by long but explicit formulas here: www.lmfdb.org/knowledge/show/ec.discriminant
@michael4192
Жыл бұрын
Why are those coeficients important?
@PeakMathLandscape
Жыл бұрын
L-functions are important because many many hard problems in number theory, which are not about L-functions to begin with, end up being expressed in terms of L-functions and their general underlying patterns. You see an indication of this in the fact that two out of seven Millennium problems are directly about L-functions, and a third Millennium is connected to the same general sphere of ideas.
@gum8888
3 ай бұрын
And i still dont know any motivation behind
@HanniSoftware
Жыл бұрын
I’m so happy I’ve found this channel , it’s amazing math content. I hope you keep going
@dcterr1
Жыл бұрын
Wow, what an amazing video! I received a PhD in algebraic number theory from UC Berkeley in 1997, but I never understood much of what you discuss in this video before, but now I do! Excellent lecture! Now I want to study L-functions in more detail!
@denunciaBillions
21 күн бұрын
I wish I understood. Still I see no sense in all these computations I have watched this video several times and from "The LMFDB" section onwards it is just alien, the computations seem so arbirary, like for somebody who already knows what is this about, I see no beauty in this procedures and that's a hint that something is wrong. Perhaps Langlads program is just a mislead. I do not know. For example there is no clue of what does the x^2+1 equation has to be with the motivic L-function and the sequences, it is not mentioned anywhere. Why number theory is always so frustrating and failed?
@m9l0m6nmelkior7
Жыл бұрын
honestly this seems interesting but- at every step I'm like "why ?? what's the link ?" because every time you present a new way of computing L-functions I can't understand how it works and for all I know this could be a gigantic prank - really I like your tone, the animations and how you present things in your videos, the only problem is that everything you say seems arbitrary, I would love if you were to explain a bit more how you get from one object to the other and WHY it works that way !!
@PeakMathLandscape
Жыл бұрын
Would have been an epic prank!! I totally understand that many things here may seem arbitrary. But all these "why" questions have really good answers, some of which are not easy to give in a short KZitem comment. Still, at least I can give some hints. The first method of computing the L-function of K is a special case of the general procedure of "counting ideals in the ring of integers". This is the standard definition of so-called Dedekind zeta functions (check this topic on Wikipedia if you like). The second method of computing the L-function of E is more general, in the sense that the underlying principle works for any equation, including K (although in much more complicated cases we need to start by counting solutions not only "mod p" but also in "finite fields", a topic we will talk more about in future Episodes, and there is also additional complications for the bad primes). A key idea for deeper understanding is that Step 1 and Step 2 in the video corresponds precisely to computing the Euler factor for the prime number p. This is also something we will come back to. So, if you keep following the upcoming Episodes, some of the "why" questions will be resolved, and whenever you feel that's not the case, just keep asking specific questions.
@m9l0m6nmelkior7
Жыл бұрын
@@PeakMathLandscape Thank you very much ! I'll take the advice and go check Wikipedia 😊your comment already helps understanding the idea behind the steps you took in the video ! Speaking of specific questions : what is the general definition of a motive ? Is it some kind of multi-variable polynomial equation ?
@PeakMathLandscape
Жыл бұрын
@@m9l0m6nmelkior7 Now you're asking one of the hardest questions in all of mathematics! There are many rigorous definitions of the word "motive", but I think it's fair to say that a full understanding of the "true" definition is lacking. Finding this definition was one of the ultimate dreams of perhaps the greatest mathematician of the past century, Alexander Grothendieck. The most basic idea is that any system of polynomial equations define a "space of solutions", aka variety, or scheme. And this space should be decomposed into "atoms", called motives, related to the cohomology groups of the space. I'll just post a few links below to explanations of various length and depth.
@PeakMathLandscape
Жыл бұрын
Short intro by Mazur: www.ams.org/notices/200410/what-is.pdf
@PeakMathLandscape
Жыл бұрын
Longer intro by Milne: www.jmilne.org/math/xnotes/MOT.pdf
@PeakMathLandscape
Жыл бұрын
Thanks for all the comments and encouragement! This is just the beginning. To answer some recurring questions: (1) All these Episodes will become openly available here at this KZitem channel. (2) We as a team are very much in a learning process when it comes to making math videos, but as we get up to speed, the aim is to post a new video every two or three weeks, with exceptions for things like holidays. We are also building an online community around L-function exploration (and later, exploration of the mathematical landscape as a whole). This course/exploration community is subscription-based, hosted at peakmath.org, and there you will find a place for asking questions about L-functions, hanging out and discussing the course content with us and with fellow explorers. We also have challenge problems and extended written course notes for each Episode, designed to take you from the basics of L-functions to a point where you can explore the immortal problems on your own. And you will get the videos a few weeks before they appear on KZitem.
@mykolanikolayev1714
8 ай бұрын
I see a pattern on 21:46 It's like clock CCW 1 - 3pm 0 - 12 am -1 - 9pm 0 - 6 pm
@mykolanikolayev1714
8 ай бұрын
On 2:59 On place #24 (if count from 0) is the first 3. I think 24 here is connected to Ramanujan work.
@denunciaBillions
21 күн бұрын
Our research should be led by beauty. I feel number theory has been a mess for a long time. I suspect Langlands program ideas are not intuitive, which hints that we may be on the wrong path. I have opened many books on number theory and strongly feel that we are not doing things right. I think we should go back to the Eratosthenes sieve and start all over again. Perhaps we are missing a prime number; perhaps we should regard $-1$ as a prime number, for example, or find another new number like irrationals or $i$. I am very grateful to you guys for trying to explain these ideas to us, but there is something missing. There is something that makes it so frustrating, awful, and arbitrary. Please do not take it personally; rather, I see in you the openness to express all these feelings I have had for a long time. I just feel we are going in the wrong direction in number theory, and moreover, the Millennium Prizes bias distracts us from the right path. I wish there were no awards so we could get our hands back on the path Riemann started for us. Why isn't everything as we expected and wanted? Why is this theory so cursed? Why can't we just stop seeking the distribution of prime numbers and embrace the mystery? Bernoulli, Euler, Gauss, Riemann, Dirichlet, Ramanujan, Hardy... none of them could make it. We have already been warned by Littlewood when he wrote to Hardy about Ramanujan: 'It is not surprising that he would have been [misled], unsuspicious as he presumably is of the diabolical malice inherent in the primes.' How much frustration, how much have we cried, seeking to understand without getting even closer, with head in hand, losing our time while understanding we have to let it go! It seems like this thing is like the three classical Greek problems or the solution in radicals to the 5+ degree polynomial: how many lifes wasted, accidentally finding lateral results that were not intended, still not sufficient, how many souls learning to give up, how many hours walking in circles, trapped, struggling in vain... until someday we will have to accept it is not possible. Someday somebody will have the courage to prove once and for all that we cannot, that primes will hide forever, that the Riemann Hypothesis is not only futile but an axiom - even true, but an act of faith. I just can't stop, I wish I could give up, but I can't! Thank you again for hearing me. Somehow, some day, we will understand.
@yoannmery
Жыл бұрын
Glad I did not spend too much time trying to guess the sequences at the beginning 😅
@andyl.5998
Жыл бұрын
20:35 The construction of K made me rewatch 3b1b's "Pi hiding in prime regularities" video. I couldn't make much sense of that video the first time, but it's a pleasant surprise to meet gaussian integers and χ function again in this video.
@pourtoukist
Жыл бұрын
The magic of KZitem lies in this kind of videos. Having access to this knowledge and this quality of content for free is simply amazing!
@sahhaf1234
7 ай бұрын
Totally agree...
@EdwinSteiner
Жыл бұрын
I can hardly believe that you gained such a large audience with just two videos! It's well-deserved since the videos are very good and much needed. Automorphic forms and their connections seem to be very beautiful subjects but almost all lectures one can find online are just atrociously bad and the material on Wikipedia isn't great either on these topics. These areas seem to attract people who are bad at pedagogy or disinterested in it. You are a welcome exception. I wish I knew how to attract so many views since I make videos on quantum mechanics and it seems really difficult to get any audience for longer videos on serious topics at more than a basic level.
@PeakMathLandscape
Жыл бұрын
Hi Edwin, thanks for the kind words. Mysterious are the ways of the Algorithm. Your videos seem very well put together, and I wish you luck!
@LUCASTAVARESCARDOSO
Жыл бұрын
As someone who was looking for a clear exposition of the main ideas in the Langlands Program, you were the perfect match. Thank you so much doing that. As a mathematical physicist, I usually try to bring tools from distinct areas of math to understand and deeper explore a model or theory which originally had nothing to do with what I brought in, and when I succeed in making interesting connections, it exhilarates me profoundly. Lately, I have also come across a family of problems which I think the ideas in the LP apply perfectly. Please, continue your wounderful work.
@RandomBurfness
Жыл бұрын
I'm sorry I may not have paid attention but, what is a "bad prime"? There are so many things that would need definitions.
@PeakMathLandscape
Жыл бұрын
Good point. Let me just answer the bad primes issue for now. For equations like K and like E, there is an explicit formula that lets you compute a quantity called the discriminant. The most well-known case is for ax^2+bx+c=0, where the discriminant is b^2-4ac. The idea then is that bad primes are the primes which appear as divisors of the discriminant. So for x^2+1=0, by the previous formula, the discriminant is -4, and the only bad prime is 2. The general formula in the case of E is equally explicit, but a bit involved, and it gives you that the discriminant is -28. So the bad primes are 2 and 7 for this equation. If you want to actually see the general formula, go to this webpage and click the word "Discriminant" under the heading "Invariants": www.lmfdb.org/EllipticCurve/Q/14/a/5
@cblpu5575
Жыл бұрын
Incredible. I hope you go as far in depth as possible
@piotr4917
Жыл бұрын
Hi PeakMath. Thank you for the series and for such broad view on the subject. I am waiting for each episode. I hope that you will have time and motivation to finish the saga - all
@piotr4917
Жыл бұрын
All the episodes and all the season. A small coincidence - I own the book presented in this episode for several years now - after watching interview with the author on Numberphile :)
@quixoteyeah
Жыл бұрын
I'm barely hanging on, but these are great videos. Really well done. Thanks.
@tanchienhao
Жыл бұрын
This is an amazing series I can’t wait to see where it’ll lead us to
@welcomeblack
Жыл бұрын
You've managed to make a 50 minute video without really saying anything at all. How is counting the number of norm appearances in the grid of integers at all related to x^2 + 1 = 0? Or why are some primes bad and some good? If you don't explain the connections then it all just feels like random disjoint games we're playing with numbers/geometry/sequences/combinatorics that you've strung together to give a particular sequence. So now, at the end of the video, I'm just left with the impression that the Langlands Program is just about noticing that some games of number shuffling are related to other games of number shuffling, and that we have no idea why.
@elieabourjeily8520
Жыл бұрын
I'm so happy I've found this channel. It's amazing math content. I hope you keep going.
@Npvsp
Жыл бұрын
You know what I believe? I believe that you did prove Riemann Hypothesis, and that you are so certain of your proof that you decided to walk us though this magnificent videos under a form of Saga, giving us the basics and then the advanced tools to understand the proof you derived, and the last video will be like “and here is the proof of RH”.
@PeakMathLandscape
Жыл бұрын
I wish this was true!! 🙂
@lioncaptive
Жыл бұрын
Agree with you 🎉
@azzteke
Жыл бұрын
No.
@fibbooo1123
Жыл бұрын
Quadratic reciprocity is one of the few topics that I've tried to learn a dozen times and each time have come away without having gained anything. I wish myself luck fo this series!
@oliverlong345
Жыл бұрын
Yes! I'm in the same spot. I really hope that this series is more enlightening once the technical details start. Especially since it seems to approach quadratic reciprocity as part of a whole family of reciprocities.
@prarobinson
Жыл бұрын
Thank you very much for posting this. I love all the connecting, but what motivates counting norms of the Gaussian Integers? Seems to come out of the blue. Ah! We get there at minute 47!
@djsmeguk
Жыл бұрын
This is fascinating stuff. Where does zeta itself sit in all this? It's used to generate a bunch of stuff but does it have the alternative forms you describe? What do they look like? What's it's automorphic representation? I hope you explore this soon. Loving the series!
@PeakMathLandscape
Жыл бұрын
Where does zeta sit - very good question. As a motivic L-function, it comes from the equation x=0, which we geometrically think of as a point, or the "unit motive". No mystery really. But on the automorphic side, there is a rather significant mystery, as the Riemann zeta is associated BOTH to the "trivial automorphic representation" AND (in a different way) to the Jacobi theta function, which is not trivial in any reasonable sense. In fact, this tension on the automorphic side may also be a clue in the search for F1-geometry. If you find this interesting, you can find further hints in this MathOverflow discussion: mathoverflow.net/questions/265584/is-there-a-langlands-philosophy-reason-for-the-fact-that-the-l-function-of-the
@kusy
Жыл бұрын
The background music helps everything to sink in
@TheSummoner
Жыл бұрын
42:42 Isn't the product for the generating function f(q) missing a factor of q before the product symbol?
@PeakMathLandscape
Жыл бұрын
Good point, that is absolutely the standard convention in the context of modular forms. We were sneaking up to the generating series through the "multiplication table", which was a way to do Cauchy inversion of the coin count sequence, but without the language of formal power series. And in the context of Cauchy inversion, it is more natural to think of the leading element 1 as the constant term rather than the degree 1 term of the power series. So that's why we didn't include the q in front. In retrospect one can certainly debate whether this was the best choice or not...
@TheSummoner
Жыл бұрын
@@PeakMathLandscapeI see, I was just confused for a bit because the product you gave for f(q) causes the coefficients of the series to be shifted to the left of one place with respect to the indexing you gave previously in the video.
@palfers1
Жыл бұрын
I'm sorry, but you have not explained how x^2 + 1 = 0 has anything to do with counting Norms. What's going on here?
@PeakMathLandscape
Жыл бұрын
Here is one way of thinking about this connection, in three steps. The first step is that x^2+1=0 is an equation that "generates" the Gaussian integers from the usual integers. The second step is to explain where the notion of norm comes from. It comes from algebraic number theory, where the norm is one of the most basic and natural invariants of an algebraic number. The third step then is that it's natural to ask how many Gaussian integers there are of a given norm, and this counting problem gives us the L-function (in this case called the Dedekind zeta function). I think the first and third step were explained in the video, but not the motivation for the second step.
@baticadavinci3984
Жыл бұрын
Please don't make us wait another 10 years for the next episode. I don't know a lot of math but I still enjoyed this stuff!
@suzy6091
4 ай бұрын
Oh my god . These lectures are just pure gold. Thank you so much 😊👍
@abdelilahtamesna6981
Жыл бұрын
About time lol
@adityakhanna113
Жыл бұрын
This is amazing. The quality and level of content is great even for working mathematicians. I do have a request, if you could have homework problems or things to work on by ourselves? That would be nice!
@PeakMathLandscape
Жыл бұрын
We have lots of such problems in the course notes. Currently these are available only for community members at peakmath.org, so check out the FAQ there to begin with, and then think about whether joining the course would be of value to you.
@minamagdy4126
2 ай бұрын
The K function looks incredibly like the formula for pi/4 (sorry, not good with names) made by reciprocals: pi/4 = Sum(n>=0) (-1)^n/(2n+1) In fact, replacing the numerators by zeta function applications seems to be a definition of that L function: L_K(s) = Sum(n>=0) (-1)^n z((2n+1) s)/(2n+1)
@DestroManiak
Жыл бұрын
Amazing series. Finally someone actually getting into some details in an accessible way. I didnt quite understand where 1, 2, 7, 14 came from, however. Something to do with the "bad primes"? But why 14, but not 14^2 etc?
@PeakMathLandscape
Жыл бұрын
@DestroManiak Good question. Yes, 2 and 7 correspond to the bad primes. Also, the number 24 is important in the theory of modular forms, and 1+2+7+14 happens to be precisely 24. Of course, these are just two interesting connections and not at all an explanation. A longer explanation would involve some theory of the special subclass of modular forms called "eta products" or the more general notion "eta quotient", which you can google. Just for inspiration, here are slides for a talk on the number 24 by the legendary John Baez: math.ucr.edu/home/baez/numbers/24.pdf
@Mr.Loewenzahn
25 күн бұрын
Im absolutely blasted by this channel. Thank you very much for this amazing content
@realdarthplagueis
Ай бұрын
Excellent videos! Just discovered your channel, and this is the BEST I have seen on this topic.
@jackr1734
2 ай бұрын
Can i have this answered...?...are imaginary numbers a representation of 3d coordenates relative to a 2d plane?
@catoyuvone8475
Жыл бұрын
:D
@danielemariettabersana595
Жыл бұрын
Hi. Thank you for these really good and useful introductory lectures!! I have a couple of questions here. 1st question... Consider the eq. E, I understood thecrecursion rule to get the Dirichlet coefficient for 3^2=9 but what about 3^3=27 that should have to be 4? 2nd question... For all the powers of the "bad prime" 7 we always have 1... What is the rule here, being apparently different from the one used for the one used for the number 2. Thank you! Kind Regards. DMB
@danielemariettabersana595
Жыл бұрын
I'm back... I undertood my question nr. 2: it is (-1)^n for the prime 2 and 1^n for the prime 7, but I have a new question: 18 = 2x3^2 and the multiplicative rule gives -1x(-2)^2=-4 instead of -1. Why? Thank you again. Regards. DMB
@PeakMathLandscape
Жыл бұрын
Quick answer: The multiplicative rule for 18 = 2 x 9 means that you get the coefficient a18 from the product a2 times a9. So you cannot use a3 here at all. In other words, you must decompose into prime powers, not primes. Does this make sense?
@danielemariettabersana595
Жыл бұрын
Clear now to me! Thank you! And what about the 27?
@PeakMathLandscape
Жыл бұрын
@@danielemariettabersana595 How I wish we had a blackboard instead of YT comments! But ok, let's do the general principle. On the LMFDB page for the L-function, you will find the Euler factors in a table, and for the prime p=3, I find in the table that the Euler factor is 1+2T+3T^2. Now there are two equivalent ways of saying what the rule is for the sequence a1, a3, a9, a27, a81, etc, indexed by powers of 3. First way: The generating series a1 + a3 T + a9 T^2 + a27 T^3 ... is equal to 1/(1+2T+3T^2). Second way: The sequence a1, a3, a9, a27, a81, ... is linearly recursive, where the recursion rule says that each element in the list can be obtained as "minus 2 times the previous one, minus 3 times the previous one before that". For example: a27 is given by -2*a9 - 3*a3. I also want to emphasize that the idea of Euler factors is explained in more detail in Episode 4. For reference, here is the LMFDB page: www.lmfdb.org/L/2/14/1.1/c1/0/0
@danielemariettabersana595
Жыл бұрын
Very useful and clear! Thank you again! Best Regards. DMB
@davidsunderland8063
9 ай бұрын
8 9 L in regards to people's response to Ants wants compared to humans wants from "'God"' 🤔 Or the movement of ants
@hdd_1230
Жыл бұрын
unpaid side hustle
@beanmanlicklick1272
4 күн бұрын
Really showed me how much I've yet to learn on the beauty of math
@cdenn016
Жыл бұрын
If only math books were written this way. I might actually learn something
@wilderuhl3450
Жыл бұрын
This video is gonna require a 4th or 5th watch. Good stuff
@gianlaager1662
Жыл бұрын
Great Video so far ( 35:06 ), but one question remains what maniac came up with that algorithm.
@suzy6091
4 ай бұрын
I think it would have been better if the name of the lecturer and their designation was mentioned. Please do it future
@Thats_A_Dummy_Name
Жыл бұрын
I like your videos. From this one I miss the connection to the other part. Why the fuck are we now doing this?
@dontwannabefound
5 ай бұрын
Why did you stop producing, keep making these videos, it is our nerd pr0n😊
@VolkGreg
Жыл бұрын
For x²+1=0, I calculated the sequence defining D(s) = sin (πs/2) OR 1,0,-1,0,... L(s) = Σ D(s/n) for n=1..s and s/n integer Examples L(25) = D(25) + D(5) + D(1) = 1+1+1 = 3 L(28) = D(28) + D(14) + D(7) + D(4) +D(2) + D(1) = 0+0-1+0+0+1 = 0 L(29) = D(29) + D(1) = 1+1 = 2
@PeakMathLandscape
Жыл бұрын
Very nice!!
@jermeekable
Жыл бұрын
is sage python code? ik on aws sage is like jupyter notebooks but this seems different
@OblateBede
8 ай бұрын
This is an interesting video series. I am curious to see where you are going with this. That being said, I find most of it unmotivated and not rigorous. Please do not misunderstand; this is not necessarily a criticism, it is more like a request for more information. I have a fairly extensive knowledge of abstract mathematics. I will not give the details of my background here, but let it suffice to say that one could write what I know of analytic number theory on the back of a postage stamp; nevertheless, I find it interesting. Some questions (or observations): I take it that an L-function is a meromorphic function on some region of ℂ, given by some power series, which is then analytically continued to all of ℂ (minus the poles). Then the power series, called the Dirichlet series, is somehow the “motivic” form, although I do not understand precisely what is meant by a motive. Perhaps the L-functions coming from Galois representations are a bit easier to understand? In this case, I am thinking of the Euler products of the characteristic polynomials of the Frobenius maps. I am completely at a loss to understand the automorphic L-functions. I suppose that my question is really this: is there something I could read, a book or an article, that would give me a quick roundup of the theory?
@PeakMathLandscape
8 ай бұрын
In Episode 4 we walk through a really nice paper of Farmer, Pitale, Ryan and Schmidt (Analytic L-functions: Definitions, Theorems, and Connections). Another option is this survey article of Cohen: www.numdam.org/item/10.5802/jtnb.920.pdf
@lietpi
Жыл бұрын
Amazing content. You've managed to make these concepts very accessible and interesting. Thanks.
@chrissch.9254
Жыл бұрын
Another amazing video - that‘s why I love Number Theory so much…
@michaelschnell5633
Жыл бұрын
All your examples of L functions have whole numbers as Dirichlet coefficients. Why is this ? Does the general definition of L functions not allow for real or complex coefficients ?
@PeakMathLandscape
Жыл бұрын
You are right! In our examples the Dirichlet coefficients are integers only because we picked four relatively simple examples.
@XrcyhikUbhdfbjdf
23 күн бұрын
Garcia Patricia Anderson Jason Robinson Frank
@letitiabeausoleil4025
Жыл бұрын
The bracket (x+y+1) is never 0. kzitem.info/news/bejne/lZiwuXqEfJt8paA
@PeakMathLandscape
Жыл бұрын
It sometimes is, because we work mod p. So for example (2+0+1) is equal to 0, when we work mod 3.
@letitiabeausoleil4025
Жыл бұрын
@@PeakMathLandscape Why is the "mod p" idea applied to the bracket?
@PeakMathLandscape
Жыл бұрын
@@letitiabeausoleil4025 In the setting seen in the video, the entire computation is done "mod p", so the idea is applied to all expressions including the bracket. Does this make sense? Or are you asking for a different, deeper kind of "why"?
@letitiabeausoleil4025
Жыл бұрын
@@PeakMathLandscape Would we "mod p" before or after * in calculating x * x or both times?
@PeakMathLandscape
Жыл бұрын
After. In addition, we could also do it before, but that would not change the end result.
@stevencraighead3808
5 ай бұрын
Where can I find your Sagemath code?
@electra_
Жыл бұрын
while this topic seems like it could be interesting, the way you explain it does not help me grasp any intuition of what is happening. Overall, you seem to present examples of various patterns, but with no overarching rules that tie the patterns together, it just feels like "what's the pattern? well... if you do these very specific rules, you get the pattern. and if you do this totally different set of rules, you get this other pattern." What's missing is some motivation or proof of where any of these rules come from or how they relate to each other. While I'm sure there is a greater logic to how this works, from how it is explained there's really no way to know these aren't just some random patterns from math that you chose and decided to call L-functions, given the lack of a rigorous definition for them.
@electra_
Жыл бұрын
it definitely seems like a topic that's extremely complex and probably hard to follow along with if you start by providing more rigorous definitions, but it just seems that without any concrete knowledge it's hard to really see where it's going.
@liangyuaq-qoyunlu407
Жыл бұрын
you are supposed to have a basic understanding of the langlands program before watch this
@jackr1734
2 ай бұрын
We need to complete reimann's spiral :'(
@rtfgx
Жыл бұрын
Where did the 2 7 and 14 come from in the product?
@jeffreyhowarth7850
6 ай бұрын
Counting coins with bad primes of an elliptic curve gives you the L-function of that elliptic curve. So elliptic curves with the same bad primes have the same L-function?
@PeakMathLandscape
5 ай бұрын
No, you can certainly have different elliptic curves with the same bad primes and different L-functions. For an example, take the curves with label 26.a1 and 26.b1 in the LMFDB.
@PeakMathLandscape
5 ай бұрын
www.lmfdb.org/EllipticCurve/Q/?conductor=1-99
@gaussniwre866
Жыл бұрын
What is this app you are using on iPad? Seem ideal for notes on PDFs and diagrams...
@PeakMathLandscape
Жыл бұрын
Notability
@miguelcerna7406
11 ай бұрын
30:48 'Bad Primes' .... Why are they considered 'bad'? What determines a bad prime?
@kylebrown2160
10 ай бұрын
I have the same question - he's skipping all this stuff with no explanation for the "basic calculation" he's doing. If the calculation is basic then just say what it is you're doing. I keep feeling like I've missed something in the video.
@manelvidiella8004
Жыл бұрын
Thanks for yur amazing job!!!
@reapingshadow2866
Жыл бұрын
Good news the story continues! :D
@thomashoffmann8857
Жыл бұрын
Around 33:00 : why is 12 factored as 3x4 and not 2x6? Which factorization is favored when it's not unique? 🤔
@PeakMathLandscape
Жыл бұрын
Great question. The idea is to factorize into prime powers, like 3 (i.e. 3^1) or 4 (i.e. 2^2), or 625 (i.e. 5^4). And 6 is not a prime power.
@thomashoffmann8857
Жыл бұрын
@@PeakMathLandscape thank you 🙂 Makes sense now 👍
@kenkiarie
Жыл бұрын
Hallelujah! Thank you sir!
@RainerNase-b3q
Жыл бұрын
Will it be possible to ask questions as of member of peakmath that are not allowed here?
@PeakMathLandscape
Жыл бұрын
For members (at peakmath.org) we make a promise to answer all questions, and we could also have longer discussions back and forth about more complex mathematical problems, things that are unclear, recommendations for reading, etc. Like in a classroom :-) For questions here in KZitem comments, just ask away anything you like, and we will try to answer to the extent we have time. No promises though. Already for E1 I think there were more comments than we have had time to read.
@RainerNase-b3q
Жыл бұрын
Thanks, I think I will use the "peek" offer as I do not know, if my time budget allows to follow in more detail. I had raised a question here, that vanished, but it referred to another video kzitem.info/news/bejne/sIlovGp8hXieq4o, so that could be a valid reason ;-) The question was: if, instead going forward or backward for square or non square vectors, why not going backward for all and the forward twice for every square. The question at it's core is about communication: firstly I was surprised about the result of reaching sqrt(P) and the fact, that the point is on the axis, then I thought: if I immediately see this second way: just start from zero and forget about all the non-squares, why didn't he show it. It's impossible, not to be aware of it. So why ;-) As the topic of this RH Saga is to make obvious, what is not obvious yet.
@Perplaxus
10 ай бұрын
this is so high quality. Very good
@ajaikapoor7721
27 күн бұрын
wow! very well presented
@ka_1242
Жыл бұрын
Keep going, looking forward to it!
@양익서-g8j
4 ай бұрын
어차피 될 일은 되고 풀일 문제는 풀린다.
@asnierkishcowboy
Жыл бұрын
Great video. I never thought about how a motiv "is" a L-function and how this "is" a prime. But its obvious in the simple cases. Take the projective quadric defined by X^2 - p Y^2 = 0 over the rational numbers for example. Its Chow motive is indecomposable if the squareroot of p is not contained in the rational numbers. This is true for example if p is a prime number. Now lets adjoin some numbers to the rational numbers. For example the squareroot of p. We obtain a field Q(sqrt(p)) The Chow motive now decomposes as two Tate (or Lefschetz) motives. The prime number p now becomes decomposable into sqrt(p)*sqrt(p) in the ring of integers of Q(sqrt(p)). I have no idea what "happens" to the L-functions in these cases or how this reflects in the L-functions. Btw, the same works also if one considers the affine quadric given by X^2 - p = 0. Motives of affine varieties are less understand than projective ones in general. Of course theres also other motives than Chow, like Nori motives.
@elias_toivanen
Жыл бұрын
We want more...we want more :-P
@dmitrisurchis
Жыл бұрын
Thank you guys. That's Mathemagica! I would be convinced, but BAD prime numbers! That's an Epic!
@peppybocan
Жыл бұрын
This looks amazing, but the list of prerequisites to understand the theory (just skimming through the Emerton's document) includes soo much advanced mathematics that it really puts me off :D I am just a poor software engineer for a love for cryptography.
@michaelschnell5633
Жыл бұрын
This is unbelievably interesting stuff ! Thanks for the video. A dummy's question: (In the third video) You provide a complicated formula for the analytic continued Zeta function. I seem to remember that an analytic function can be written as an infinite grade polynomial, and hence the analytic continuation should be the polynomial that is equal to the original function in the defined range of same. Does it not make sense to give a formula for the coefficients of that polynomial as the definition for the analytically continued function ?
@alexpotts6520
Жыл бұрын
Any sufficiently advanced mathematics is indistinguishable from magic
@mariotabali2603
Жыл бұрын
This is fkngng gold Mine.
@lindavid1975
Жыл бұрын
To me, all primes are bad.
@RainerNase-b3q
Жыл бұрын
Isn't it that geometry and algebra also are dual (0:59) and did Descartes showed this by the introduction of coordinates? Reading one of his books I was surprised how to solve equations just by geometry. At 14:37 the gaussian integers are introduced. There is a rule like: it is a set of numbers with two parameters where every parameter is starting from zero and stepped by one. But it could be more difficult, like: search the solutions I, J, K, integers, of f (x, y) = x² +y² where f (I, J) = f (K, 0). Now we look to find integer solutions of a function over R², not the Pythagorean triples. And from these points we define the larges grid that contains all of those points.
@venaterox867
Жыл бұрын
26:10. I don’t understand
@Driancreid
Жыл бұрын
This is a great video, particularly because you keep forging ahead without explaining or justifying every point with complete rigour. In this way you can get more quickly to the interesting points before you lose the watcher. Well done from someone who specialises in Maths Education and Online Learning!
@Salmanul_
9 ай бұрын
Great content!
@illogicmath
Жыл бұрын
After the prelude of Bach Cello Suite #1 that he used as background for the Sage calculation I got completely lost
@corneliusgoh
Жыл бұрын
I read in 2014 Ed. Frankel's book "Love of Maths " the very page 88 which I was totally lost... Now after this video, I understand 90% ...thank you for the great video.
@alexakalennon
Жыл бұрын
I just watched a few lectures on the langland program Now this amazing second video. I m baffled. Thank you for this great efforts
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