L-functions and the Langlands program (RH Saga S1E2)

[@amritawasthi7030]

Top tier content

[@NachoSchips]

PeakMath Content you might say

[@jackr1734]

For the average alien

[@PeakMathLandscape]

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 KZbin 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 KZbin.

[@mykolanikolayev1714]

I see a pattern on 21:46 It's like clock CCW 1 - 3pm 0 - 12 am -1 - 9pm 0 - 6 pm

[@mykolanikolayev1714]

On 2:59 On place #24 (if count from 0) is the first 3. I think 24 here is connected to Ramanujan work.

[@LordDragour]

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.

[@dcterr1]

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!

[@LordDragour]

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?

[@pourtoukist]

The magic of KZbin lies in this kind of videos. Having access to this knowledge and this quality of content for free is simply amazing!

[@sahhaf1234]

Totally agree...

[@navjot5445]

I have a gut feeling that you have proved RH is true, and gonna shock the whole world in the last episode 😅

[@quarkonium3795]

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]

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

[@yoannmery]

Glad I did not spend too much time trying to guess the sequences at the beginning 😅

[@LUCASTAVARESCARDOSO]

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.

[@HanniSoftware]

I’m so happy I’ve found this channel , it’s amazing math content. I hope you keep going

[@andyl.5998]

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.

[@cblpu5575]

Incredible. I hope you go as far in depth as possible

[@piotr4917]

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]

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 :)

[@fibbooo1123]

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]

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.

[@tanchienhao]

This is an amazing series I can’t wait to see where it’ll lead us to

[@suzy6091]

Oh my god . These lectures are just pure gold. Thank you so much 😊👍

[@EdwinSteiner]

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]

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!

[@quixoteyeah]

I'm barely hanging on, but these are great videos. Really well done. Thanks.

[@baticadavinci3984]

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!

[@TranSylvainie]

Perfection ! Can't wait for the adventure to continue !

[@Driancreid]

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!

[@RSLT]

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!

[@Mr.Loewenzahn]

Im absolutely blasted by this channel. Thank you very much for this amazing content

[@m9l0m6nmelkior7]

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]

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 KZbin 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]

@@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]

@@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]

Short intro by Mazur: www.ams.org/notices/200410/what-is.pdf

[@PeakMathLandscape]

Longer intro by Milne: www.jmilne.org/math/xnotes/MOT.pdf

[@rohitchatterjee2327]

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

[@realdarthplagueis]

Excellent videos! Just discovered your channel, and this is the BEST I have seen on this topic.

[@lietpi]

Amazing content. You've managed to make these concepts very accessible and interesting. Thanks.

[@Perplaxus]

this is so high quality. Very good

[@ankittripathi3851]

You are too serious man, a smile every few moments would do the trick !

[@AnnevanRossum]

Very grateful for this series. Thank you so much! 🤗

[@kusy]

The background music helps everything to sink in

[@darkseid856]

This is gold . Thank you so much for providing such high level content ❤

[@Npvsp]

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]

I wish this was true!! 🙂

[@lioncaptive]

Agree with you 🎉

[@azzteke]

No.

[@eldermorph2023]

brilliant series so far. just what i was looking for. please keep it up

[@beanmanlicklick1272]

Really showed me how much I've yet to learn on the beauty of math

[@modernsolutions6631]

I hope this leaves all the attention it deserves

[@wilderuhl3450]

This video is gonna require a 4th or 5th watch. Good stuff

[@Tomyato]

As a mathemician, this channel is a gem, どうも有難うございます　😄

[@alexakalennon]

I just watched a few lectures on the langland program Now this amazing second video. I m baffled. Thank you for this great efforts

[@chrissch.9254]

Another amazing video - that‘s why I love Number Theory so much…

[@saulmendoza1652]

This is like Sesame Street... Colorful and entertaining... And also top tier

[@adityakhanna113]

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]

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.

[@cosmicwakes6443]

Such a beautiful presentation on these mysterious objects. Thank you.

[@psycdalex]

This is incredible. Thank you for gifting this to me

[@DouglasKubler]

Great teaser journey. I hope the next step follows soon so my misunderstandings can be explained to me. For anyone who doesn't like dealing with viewing papers on screens or downloading and printing same do as I have done - In case I remain stuck on a deserted island I have brought a copy of edited by Julia Mueller and Freydoon Shahidi Chapter 15 is the recommended paper by MATTHEW EMERTON

[@PeakMathLandscape]

That's the one item you really need on a deserted island!

[@ka_1242]

Keep going, looking forward to it!

[@samdailley8067]

I am just this second about to watch the video and I am just as excited as I am to watch this vs Oppenheimer :)

[@corneliusgoh]

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.

[@Craftesteur]

Incredible work on incredibly hard stuff!!! Much love from France

[@kenkiarie]

Hallelujah! Thank you sir!

[@chriscopeman8820]

I can follow the steps but the motivation for doing the steps eludes me.

[@jcg9130]

What a pleasure to discover this content!

[@djsmeguk]

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]

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

[@prarobinson]

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!

[@ΙσίδωροςΤσαούσης]

What an amazingly intuitive concept

[@reapingshadow2866]

Good news the story continues! :D

[@RandomBurfness]

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]

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

[@dmitrisurchis]

Thank you guys. That's Mathemagica! I would be convinced, but BAD prime numbers! That's an Epic!

[@alp2302]

Nobody says otherwise, I encourage you to pursue doing this. This kind of content is getting pretty popular I presume nowadays, and there is a load of people coming up with popular math videos. Even though I enjoy and find important those videos, they trade off something for popularity and general accessibility, that is, abandoning depth. I do not want to be misunderstood, being unnecessarily convoluted is not good for sure, and I dont think every youtube channel should be diving to the same depth. However, it is really good to notice a new level of that deepness on KZbin. Well, great job =D

[@michaelraasch5496]

Super interesting! Thanks for the video.

[@fawzibriedj4441]

Very fun watching!

[@ajaikapoor7721]

wow! very well presented

[@lcyken]

The use of Sage as a blackbox is a brilliant idea

[@MarkTimeMiles]

Beautiful work!

[@Hamboarding]

Wow! Great series!!!

[@GastroenterologyPINNs]

Great introduction!

[@DestroManiak]

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]

@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

[@matanfih]

Was waiting for it!

[@Maurdekye]

wow i understood

[@Salmanul_]

Great content!

[@iancu_de_hunedoara]

One of the best math channels.

[@manelvidiella8004]

Thanks for yur amazing job!!!

[@yoyokojo651]

Simply amazing

[@asnierkishcowboy]

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.

[@suzy6091]

I think it would have been better if the name of the lecturer and their designation was mentioned. Please do it future

[@michaelschnell5633]

Some questions regarding the Fourier series alike "cos wave" sum of -cos(ti log(x)) ... functions, that seem to approach a sum of delta peaks. Seemingly these are numerically very well examines, Supposedly by calculation for a rather high count of terms, hence those results might be considered likely: Is the function considered to approach zero everywhere but at whole numbers ? Do the peaks at primes all have the same "Energy" ? Is a formula suggested for the "energy" of the "weaker" peaks at prime powers ? Are there also (even weaker) peaks for products of (different) primes (i.e. for all whole numbers) ? Thanks for listening ! -Michael

[@PeakMathLandscape]

Excellent questions! The plan is to address some of these questions later in the course. They are related to so-called "explicit formulae" in analytic number theory, see e.g. en.wikipedia.org/wiki/Explicit_formulae_for_L-functions

[@cdenn016]

If only math books were written this way. I might actually learn something

[@minamagdy4126]

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)

[@888Xenon]

Fantastic stuff, keep it up!

[@michaelschnell5633]

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 ?

[@FreeZh-xyz]

42:36 Should the generating series be 1 over the product? Thanks

[@peppybocan]

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.

[@TheSummoner]

42:42 Isn't the product for the generating function f(q) missing a factor of q before the product symbol?

[@PeakMathLandscape]

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]

​@@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.

[@shiva_kondapalli]

Top quality content, 👌🏻 Love it! I am an aspiring mathematician pursuing a masters who really wants to understand the relation between number and shape. I am curious to know if F1 geometry relates in any way to the absolute galois group? If I am not mistaken, isn't the langlands program studying the linear part of the absolute galois group. While anabelian geometry takes into account the non-linear actions of Gal(Q_bar/Q). Can F1 geometry help in bridging langlands with anabelian phenomenon? Thanks and please keep such wonderful content coming, it is invaluable for seekers like me.

[@PeakMathLandscape]

Very good question. I know almost nothing about anabelian geometry, so cannot really speak to possible relations with Langlands etc. But maybe some loose comments, just throwing out some ideas. 1) There is a fundamental difference between entities that are seen by L-functions and entities that are not seen. Take for example (see Wikipedia) Szpiro's conjecture, which gives a relation between the discriminant and the conductor of an elliptic curve. These are two invariants, the conductor is "seen" by, or can be "recovered from" the L-function, while the discriminant is not. So we don't expect methods purely from L-functions (and here I include Langlands-related objects like motives) to be strong enough to yield something like Szpiro, but it is hoped that methods from anabelian geometry might. This is of course a very fluffy argument - maybe there is some clever way to extend L-function theory or combine it with other tools etc, so one should never say never with absolute certainty. 2) There is a significant philosophical difference between group actions on linear objects (vector spaces) and group actions on more general ("un-linear"?) objects (like sets, simplicial sets, geometric/topological/combinatorial) objects. One setting where this difference appears is in actions of Galois groups (absolute or otherwise). Another setting is in F1-geometry, at least in one of the basic intuitions that vector spaces should be replaced by sets or pointed sets. But I don't know if this parallel means that one can make a direct link between some version of F1-geometry and some Galois actions. There are lots and lots of things here one could spend time thinking about, and there are probably links, both known and unknown.

[@shiva_kondapalli]

@@PeakMathLandscape Thank you for the answer. I know very little about these ideas, but can't help thinking about such fascinating objects. I remember Smirnov mentioning in the F1 world seminar talk that F1 is non-linear mathematics, it was this non-linear word that got me thinking about Gal(Q_bar/Q). I know I am speculating far above my pay grade here, but Buium says that his theory of "Calculus over the integers" is essentially about encoding the curvature of Z into a group holQ. If i understand it correctly, it is a lie algebra which should be viewed as an infinitesimal deformation of the absolute galois group. Also, Connes and Consani construct Witt vectors in characteristic one; this connects non-commutative geometry to F1 and touches upon entropy as well. Exploring such connections must be seriously exciting. Speaking of excitement, I am excited for the Legend of Ishango starting in October. A mental landscape of math one can use to understand the unity of mathematics personally vibrates with me. Maybe the whole landscape is a big Topos!

[@guillermo3412]

This is awesome content, new subscriber. 👍

[@multicolourmadness2712]

Hey I have a quick question about 38:40. If you take convolution as group operation on a sequence, isn't the process you describe here akin to finding the inverse element to 1, 1, 3, 4, 9, 12, 23, since 1, 0, 0, 0, 0, could be considered the identity element. I'm about to go into my third year of maths at university and this is all really interesting to me, although we don't cover anything close to this complexity yet (last year was only the introduction course to things like rings and geometry). I've been doing a lot of stumbling through maths on my own time and this channel is very exciting to me, way beyond my level but you actually explain some of these big mysterious objects in maths, so thanks a lot :)

[@PeakMathLandscape]

Yes, you are exactly right! We wanted to do this but without assuming everyone knows about (Cauchy) convolution of formal power series, and the process of taking inverses with respect to this convolution.

[@OblateBede]

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]

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

[@gianlaager1662]

Great Video so far ( 35:06 ), but one question remains what maniac came up with that algorithm.

[@danielemariettabersana595]

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]

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]

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]

Clear now to me! Thank you! And what about the 27?

[@PeakMathLandscape]

@@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]

Very useful and clear! Thank you again! Best Regards. DMB

[@RainerNase-b3q]

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.

[@alexpotts6520]

Any sufficiently advanced mathematics is indistinguishable from magic

[@DavenH]

This is killing me - what's the piano melody from in the intro? I feel like I've heard it recently elsewhere, like something from Scriabin or Chopin.

[@DavenH]

Found it kzbin.info/www/bejne/baKzoWOkn9OKns0

[@cg11235]

Thanks! ❤

[@bini420]

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]

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]

@@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]

@@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.

[@thomashoffmann8857]

Around 33:00 : why is 12 factored as 3x4 and not 2x6? Which factorization is favored when it's not unique? 🤔

[@PeakMathLandscape]

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]

@@PeakMathLandscape thank you 🙂 Makes sense now 👍

[@elias_toivanen]

We want more...we want more :-P

[@benpaz9548]

we pray for S1E4 🙏

[@mariotabali2603]

This is fkngng gold Mine.

[@pal181]

I hope in the next episodes it will be explained more how all of these connect, for now it's all plucked out of thin air.

[@crjensen77]

Agreed - how did you get from x^2 + 1 = 0 to counting the norm of the Gaussian integers? If that is coming in a later episode, that's all you need to say - I will get to all of the other episodes as soon as I can! This is great content!

[@DeathSugar]

I wish there were more maps like the one in the video in public access.

[@gaussniwre866]

What is this app you are using on iPad? Seem ideal for notes on PDFs and diagrams...

[@PeakMathLandscape]

Notability

[@jackr1734]

Can i have this answered...?...are imaginary numbers a representation of 3d coordenates relative to a 2d plane?