The Generalized Riemann Hypothesis (RH Saga S1E3)

This is the third episode of the RH Saga.
We continue our journey by considering the zeros of L-functions beyond the Riemann zeta function. The location of these zeros is the subject of the generalized Riemann hypothesis. The story of these zeros also give a first indication of the general relationship between prime numbers and L-functions.
The overall aim of RH Saga Season 1 is to map the landscape of L-functions, as a foundation for future in-depth exploration of some of the most immortal math problems of all time.
This video is part of a PeakMath course. Join the journey at www.peakmath.org/
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Chapters:
00:00 - Intro
02:00 - Review of examples
05:04 - Analytic continuation
12:39 - Zeros in the critical strip
16:44 - Cosine waves
21:35 - Final remarks
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Links:
1. LMFDB
www.lmfdb.org/
2. "The Music of the Primes" on Amazon UK:
www.amazon.co.uk/Music-Primes...
---
Errata:
The integral written at approx. 11:00 has a slight mistake. Instead of the number 2, the variable s should be used inside the sine function in the integrand. Correction displayed on screen at 11:20.
---
Social:
www.peakmath.org/
#RiemannHypothesis #F1Geometry #Mathematics #PeakMath #RHSaga #Langlands

[@cheedozer7391]

Just wanted to let you know that you're reaching and inspiring several high schoolers with this series. This is excellent communication!

[@eineannanas5104]

the videos are so good pls never stop making them

[@mmoose3673]

There's so much drama in this math. Consistent suprises. It feels like we're "picking up stones on the shore of a beach" like Newton said. It's so exciting to catch some glimpse, no matter how minor, of the forefront of human knowledge. How exciting!

[@jackhgallagher]

I haven't felt this excited about finding a new math channel since I discovered 3Blue1Brown (and before that, Welch Labs' Imaginary Numbers Are Real series). Your presentation style is a beautiful mixture of personal, organized, yet unpolished, which makes it feel like this is a one-on-one office hours session with a great professor. I especially appreciated the cinematography of mixing in (1) the screen recording of Notability on the iPad, (2) an overhead camera showing you writing on the iPad, and (3) the manim graphics cleaning up the handwritten tables. I look forward to being able to say that I subscribed to PeakMath when you all had but 8,000 subscribers and 3 videos--yet I look all the more forward to the many incredible videos I'll get to watch in the future!

[@alexakalennon]

This whole series is a huge heartbeat of love. Amazing work Thank you so much

[@dinosoeren]

this is the kind of content that I can't believe I can watch for free... what a time to be alive!

[@salvadorvillarreal1643]

I'm really looking forward to the next chapter and being formally introduced to L-functions. Amazing video as always!

[@minerscale]

I get the sense that there is some deep, deep structure that we get tiny glimpses from the outside, revealing just a smidgr of its beauty. I hope I get to live to see some proper progress on these problems.

[@mattkerle81]

This is the first video I've seen on the RH that wrote down the integral form of the series and showed some examples of what the integral actually looks like, thank you! Much better than low effort pulp going on about 1 +2+3+...= -1/12 without explaining that's a completely different mathematical beast!

[@ytang3]

I was going to say, "the (in)famous -0.08333... rears its ugly head!"

[@888Xenon]

Loving this series! Keep up the excellent work :)

[@flmbray]

I'm so happy to see you labeling these as "S1E#".. because that implies there will be a S2... WOOT-WOOT!!!!

[@amritawasthi7030]

Earlier I said it's a "top tier comment" But now I wanna say that it's "Peak Math content" damn amazing.

[@EdwinSteiner]

Are the zeta functions featuring in the Weil conjectures not considered "L-functions"? I'm asking because for those zeta functions an analog of the Riemann Hypothesis has been proven. However, those functions look simpler because they can be expressed as (ratios of) finite products rather than as infinite products. Is this a defining difference?

[@PeakMathLandscape]

Great question. In this series we will use the term L-function for a specific class of functions (defined next time in Episode 4) that are "global", which means that they gather information from all prime numbers. The zeta functions featuring in the Weil conjectures are "local", in the sense that they gather information only from one prime number. Both of these notions will be made much more precise.

[@emilioferrer9706]

This is brilliant. Congratulations!

[@josh34578]

This is really fascinating, thank you. Looking forward to more.

[@nickrr5234]

Beautifully presented! Look forward to the next one.

[@TranSylvainie]

I am hooked ! Good that it comes every 3 weeks I have time to rewatch the first episodes. There is an interesting 3b1b video about visualizing the analytic continuation of the zeta function. I need to come back to the process we used to create L-functions...

[@VelAntuManthureie]

Beautiful, beautiful content. I cant wait for next episode.

[@drakebrown908]

For those wondering why the zeros of K match up with the zeros of P and A its likely due to how we obtained the coeeficients of the dirichlet series. In the first video, the way we obtained the coefficients of K corresponded to collecting the coefficients resulting from multiplying A and P(try it out). Thus if either of these are zero, the whole function K is zero. Likewise E from the last video is obtained from the "inverse" of the power series with coeficients corresponding to the coin sequence.

[@rotemperi-glass4825]

please continue, this is amazing! each time a new episode comes out, I celebrate it (:

[@zeus7914]

excellent series. provides key insights into the distribution of primes.

[@FRANKONATOR123]

Such an excellent series, well done

[@joeeeee8738]

Great improvement from las chapter on explaining the topic! Kudos and keep on!

[@bryanmcdermott5353]

Can't wait for more. Keep 'em coming.

[@miguelandrade4439]

Amazing! This explanation is brilliant, down to earth and just very well done in general!! thumbs up

[@zmaj12321]

A new great math channel! What has already been shown is fascinating, but I feel like we're only just getting started.

[@FredericoKlein]

So I am not a maths person, but you explain this subject so well that it makes me think. What if this GRH is unsolvable? There are algorithms that are undecidable in computer science, so what if this infinite integral is actually a version of the same thing? Like the 3K +1 problem. It is just impossible because the whole thing cannot be solved. How would you go about proving such a thing though? Well, again not a maths person, so I could be completely wrong here, but it seems to me there is like an underlying concept of entropy hidden inside the computation itself, like it is making a complex system, or something with emergence. Anyway, fun ride, keep making videos! (maybe do a refresher on the basics of maths, since a ton of your viewers may lack some of the basic concepts you use like automorphism)

[@Rugjoint]

loving this series !!

[@steliostoulis1875]

By far the greatest youtube math series

[@toyeshjayaswal1645]

I really like this series. I've seen the proof of PNT many times before, but it's always felt symbolic. I've never actually seen things like the peaks occurring at primes, and it has me appreciate the equations more since I can actually see something surprising going on. Thanks for making these!

[@muldermachines]

Fascinating. thankyou!

[@therealist9052]

Not a mathematician (but hope to get a math degree one day!), but I'm getting the sense that what will be needed to solve this F1-Geometry problem is a new way to define numbers based on new axioms of mathematics. It seems like that new definition would have to include elements of symmetry and reciprocity, and automorphism fundamentally baked in. Ever since I came across p-adic numbers, Galois groups, Modular Forms, Quaternions etc., I haven't been able to get the idea out of my head. Could be totally wrong, but that's the impression I've gotten. Thank you to channels like yours that inspire me to keep mathematics in my mind!!

[@Finkelthusiast]

Fantastic series. My guess for the wave corresponding to A have spikes at 1mod4 and 3mod4 alternating between plus and minus respectively.

[@RSLT]

Love it very informative thank you!

[@monsieurhics4684]

I can't wait the next one 😭

[@dev_sda]

My favorite part is always the final remarks. I love this series can’t wait for Season 2

[@AvanaVana]

Ha, I just finished reading that Du Sautoy book, which is what led me to crave _more_ investigation of the RH and related topics, including this video series!

[@zathrasyes1287]

Awesome!!! Keep on going, pls!!!!!!!!!!!

[@kees-janhermans910]

Fantastic series - I'm enthralled. Non-mathematician's questions: - You're 'deriving' a series of numbers (the enumerators on the infinite series used in a Zeta function) through a mechanism, from an equation. Can you walk that back? In other words: if I have the L function numbers, can I get the original equation back? - Are the outcomes of the Zeta function, as parametrized by any L function number series, relevant/related to the outcomes of the original equation? - I assume that the 'machines' shown so far, are all specific examples of a more generalized machine function? - Is the L function 1 -1 1 -1 1 -1 1 -1 1 -1 etc the Dirichlet Eta function? Which equation yields this L function?

[@PeakMathLandscape]

These are excellent questions. Brief answers (not complete): 1) There are many things you have to make precise before you can answer the question in fully precise language, but I think the spirit of the question should be answered by saying that in some cases, you can get the original equation back, but in general you cannot. For equations in one variable ("number fields"), there are examples of different number fields having the same L-function (search for "arithmetically equivalent number fields", e.g. the Master's thesis of Lotte van der Zalm), so you cannot in general get the number field back, BUT if you only care about equations of low degree (I think it is up to 6), then the L-function does determine the number field, so in this sense, the L-function determines the equation. For elliptic curve L-functions, there is a similar discussion, in if two elliptic curves are "isogeneous", they have the same L-function. 2) Not entirely sure what you mean by "outcomes" here, but one possible answer is that the coefficients of the L-function are always related to the number of solutions of the equation modulo prime numbers (and more generally, the number of solutions in so-called "finite fields"). 3) This is related to the previous point, and yes, there is a general machine called "point-counting in finite fields" that gives the L-function of any equation. 4) That's exactly right, the Dirichlet eta function. The relation to equations requires a long explanation. The starting point is that the Dirichlet eta and the Riemann zeta have identical Euler products except for the Euler factor at the prime p=2, where the eta has a "strange" Euler factor which is not related to an equation in the standard sense of the word "equation". But there is a lot more to say that I cannot reasonably fit in a comment.

[@tracyh5751]

wonderful!

[@jennyone8829]

Thank you 🎈

[@surrealbits]

Wow. Somehow it left me with a feeling as if the equations are acting as a lens of some shape and type that when looked under this "framework" is highlighting some properties of primes. The "P" being the simplest does not "filter" anything and hence shows all primes, analogous to the delta function in convolution. An identity of "some" operation?. The "K" is also interesting. (x^2 +1) is obviously related to 'i' and may be the 2nd power of x is leading to highlighting of factorisability of primes in 2 terms under gaussian integers. The "E" equation similarly highlights something but the pattern may not be obvious as you pointed out.

[@FractalMannequin]

Not gonna lie. I'm waiting for your new videos as much as I'm waiting for the next One Piece chapters.

[@yoyo42]

Thank you so much for this chanel !!! Question : Why do the L fonction of K disapeared in the related object of LMFDB database ? How do we acces to it now ?

[@TerryMaplePoco]

amazing

[@HadiLq]

In an hour, 80 people liked it, which means it is a very hot topic.

[@bini420]

This is amazing. I never expected the series to give so much information, a birds eye view from someone who knows the land. Question tho, how much progress is their on grh. Did anyone manage to prove that at least some percent of zeroes lie on the critical strip? Or that they lie in some region around it. I remember reading somewhere that dirichlet proved his theorem on arithmetic progressions using dirichlet L functions, so do all dirichlet l function correspond to Primes in a specific arithmetic progression? Thank you for your videos

[@madly1nl0v3]

Brian Conrey and his team, in 2011, proved that more than 41% of Zeta zeroes are on the critical line.

[@giorgiobarchiesi5003]

So if the zeroes of K are all those of A plus all those of P, does this mean that K = AP ? (possibly multiplied by a constant)

[@PeakMathLandscape]

That's exactly right. And this will also be the theme of the upcoming Episode 5!

[@wilderuhl3450]

@@PeakMathLandscape episode 5? I can barely wait for episode 4. Good stuff

[@jagatiello6900]

Also K inherits the pole of P, since A is non-zero there.

[@VaradMahashabde]

Small question, why does L-function of K skip over 2? It's also factorable as (1+i)(1-i).

[@eoghanf]

@4.02 I laughed out loud (yes, it took me a few seconds to get that joke)

[@joshuasparber156]

The prime step function tends towards more vertical steps the more primes are added. At each of the vertical portions of the step the Laplace transform of the derivative of the cumulative step function would tend toward a Dirac Delta function, a spike of infinite height and zero width at the location of the transition. The spikes would continue to get skinnier and higher as the number of primes accumulate in the step function. This is due to the change at the ultimate vertical step being a truly infinite change. Thus, this derivative function would head towards a function of Dirac Delta functions at each prime as the number of primes head towards infinity. My question is, do any of the series or grouping of these series you mention here represent this Laplace Transform derivative as the number of primes head toward infinity? The interesting thing is that the human mind can conceptualize infinity in many ways without actually being able to visualize infinity.

[@benpaz9548]

we pray for S1E4 🙏

[@tomkerruish2982]

I'm enjoying these greatly. Thank you. Some thoughts: cos(t ln x) is the real part of x^(it) for real t. I presume this will come into play at some point. It appears that the sequences a_n are the values of multiplicative functions, and so these particular L-functions can be written as Euler products. My rather limited research (Google searches), however, seems to indicate that this is not the case in general. I look forward to learning more from you on this and other points.

[@PeakMathLandscape]

Good observations, and yes, we'll come back to these points! In the next Episode, we will choose to include the Euler product as one of our axioms for L-functions, since it is expected that in a hypothetical future proof of GRH, the Euler product will be an absolutely crucial ingredient.

[@JonDornaletetxe]

🔥

[@Giraffozilla]

In the integral representation of Riemann Zeta function, why aren't the values identical? Is it just the numerical error?

[@PeakMathLandscape]

That's right. We used a specific number of trapezoids for the integral and therefore got a small numerical error. The more precision you use for the numerical integration, the better agreement you would get with the actual zeta value, and in the limit the agreement is exact.

[@user-sk4kg4hr3k]

How we know that given random set of numbers form an L-function if used as coefficients?

[@jonnyoh4731]

Is it theoretically possible that proving the non trivial zeros of the L functions of A and K lie only on the critical strip, would then in turn prove the RH? Or do we also in turn think the same principle in proving the RH would be the same principle of proving for A and K?

[@YiqiXu]

Could you please tell me the source of the poem by Kazuya Kato at the end of the video?

[@PeakMathLandscape]

It was in the special volume of Documenta Mathematica for Kato's 50'th birthday. Check this page, where the original Japanese version and the translation are both in one of the first pdf files, called Prime Numbers. www.math.uni-bielefeld.de/documenta/vol-kato/vol-kato.html

[@user-sk4kg4hr3k]

1. Why polynomial for K is not X^2 =0, where this "+1" comes from? 2. Are there LFs for 3d space or may be it's better to ask about quaternions?

[@5ham1nry32]

Rie on schedule.

[@thelostmarbles4310]

Very good... but could you please explain in very basic practical terms how you got the prime cosine waves. I am enthusiastic and a bit dumb at math... thanks.

[@PeakMathLandscape]

Hi, did you check Episode 1 and the explanation there?

[@tonyblake1593]

Hi, What is the name of the note taking app and stylus you are using to write out the math?

[@PeakMathLandscape]

The app is Notability, and the stylus is an Apple Pencil (not even sure which generation)

[@tonyblake1593]

@@PeakMathLandscape Thanks! These videos are great. The coin counting you showed in the previous video for calculating 14a5's L function coefficients. What is that combinatorics actually counting? I thought it was partitions of number but it's not (as it's a different generating function).

[@PeakMathLandscape]

I guess it is some kind of partitions but where the summands are restricted to come from the "coins", and also "coloured" so that for example 2+2 appears three times. There is probably a name for these in combinatorics, which I don't know!

[@DeathSugar]

is there a proof about sin of spectrum spikes?

[@monsieurhics4684]

🥳

[@Fine_Mouche]

4:35 : why there is no automorphic function for P : x=0 ?

[@PeakMathLandscape]

There is! Maybe this could have been clearer. The sequence 1, 1, 1, ... is the L-function associated to the simplest possible automorphic object, called the "trivial automorphic representation of GL1".

[@jesseburstrom5920]

i tooled x^2 +1 mod p and somehow found a system inside the system by writing diagrams of connections still not proven since complicated but my true love is computer science and art

[@Darrida]

The Langlands Program is a rather complex area of mathematics. I think it takes about five or six years to understand what we are talking about here.

[@user-xl1bj5mv5s]

Ты теперь мой второй кумир после ньютона

[@gazzamgazzam4371]

Which device do you use in making videos? "is it an ipad?"

[@PeakMathLandscape]

Yes, an iPad Pro

[@Hamboarding]

19:24 What about 9⁇

[@v8torque932]

Why do you talk like that