this whole world feels like relatively untapped potential for the youtube education space, almost surprised there seems to be so little digestible content about this stuff in this form, but i guess its because everything is so hand-wavy and conjectural at the moment. ive been loving the introduction nonetheless
Summation: Being interested in Primes, and believe me - done a hell of a lot of research. TH-cam as a whole - only gives a basis. These videos work though, and seriously condense everything currently available, all in nice little chunks. Have learnt a massive amount more, and love the fantastic work / research / production / complexity involved. Very very good! Cannot wait for further episodes.
@@artemetra3262 , IMHO the most accessible intro into Modern Number Theory are the three volumes "Number Theory 1-3(Translations of Mathematical Monograph)" by Kato/Kurokawa/Saito, starting from basics, going to Class Field and Iwasawa theory to a sketch of Wiles' proof of Fermats Last Theorem. Lots of numerical examples. Highly recommended.
I feel like this episode is what finally tied everything all together for me and provided a good look into the math behind the magic of L functions. Maybe it's because I have a category theory background, but thjs just finally made it click
I have a bachelor's degree in computer science, and I work as a computer programmer. But I've always had an interest in mathematics. I've been using SageMath for over 15 years to study things like tetration (iterated exponentiation, including an iteration count that is non-integer, perhaps even complex), the Riemann Hypothesis, and prime gaps (especially maximal prime gaps). But I have always been daunted by L-functions, so I've never explored beyond the Riemann zeta function. I have a fascination with the Riemann Hypothesis, but I have always felt too overwhelmed to look into the GRH. This video series is opening new possibilities to me, and this episode especially has created a desire within me to study L-functions more thoroughly. Thank you for making these videos available to the public like this!
When you said that maybe the Riemann Z function could perhaps be equivalent to the vacuum state, a shiver ran down my spine and goosebumps ran up and down my skin
Is the ⊕ operation on L-functions associative and commutative? Are there identity elements, and is the quasigroup property satisfied, or are there inverse elements? Or, are these currently all open questions in the research?
The plus operation (well done getting the actual symbol into a comment!) is well understood, and it has all the properties you could hope for (associative, commutative, identity element, inverse elements) although for the inverses you have to choose a convention where things like the Möbius mu function counts as an L-function. The Dirichlet series of the Möbius mu function is the inverse of Riemann zeta, and similarly, every L-function has it's own associated "Möbius-like" inverse. The identity element is the "trivial" L-function corresponding to the sequence 1, 0, 0, 0, 0, 0, ...
Very good video with a great motivating intro. To strengthen the analogy with quantum physics, one would for example expect some "raising" and "lowering" operations to create states from the vacuum. Are there analogs to this here, maybe among the unary operations you presented? Indeed many things about L-functions remind me of quantum systems, for example the seemingly quantized "spectral parameters", i.e. the Gamma-shifts. Viewers interested in quantum mechanics might want to check out my "Physics" playlist, where I aim to explain QM at a similar intermediate level of technicality, avoiding oversimplification.
Ah, deep question. There are general hopes for some form of "quantization" in the world of L-functions, but there are many different speculative ideas on how to make such a vague idea into rigorous mathematics. Impossible it is, to discuss all of this in TH-cam comments, but a few brief remarks. (1) Google "Berry-Keating conjecture" or "Berry conjecture" for one direction. (2) In a different direction, it feels like the real nontrivial zeroes behave like bosons, and the other nontrivial zeroes behave like fermions, in that the real zeroes can congregate into a multiple zero at the central point. Perhaps this is a nonsensical idea, perhaps it could lead somewhere. See Episodes 7 and 8 on the BSD conjecture! (3) For raising and lowering operators, there are no obvious direct analogues as far as I know, but this is certainly an idea worth playing around with. If you start with a degree 2 L-function, say L_E, and apply the symmetric power operator Sym^k for k=1, 2, 3, 4, ... you get a sequence of L-functions of degrees 2, 3, 4, 5, ... and this is one of many possible starting points for playing around. Could you define a "raising operator" on L-functions that in some generality takes Sym^k L to Sym^{k+1} L??? (4) The reason for actually comparing Riemann zeta to the vacuum is that the Riemann zeta is conjecturally the only L-function with a pole at the edge of the critical strip. All other L-functions should be entire (i.e. no poles) except if they contain Riemann zeta as direct sum component. Could this pole be given a physical interpretation as "mass zero"??? (5) As you point out, the Gamma factor shifts look a bit quantized. For full appreciation of this, one should also look at transcendental L-functions, for example from Maass forms, where these Gamma shifts are not real, and seemingly transcendental, and where they appear with a close relationship to eigenvalues of a Laplacian operator. I don't think anyone has a complete understanding of how to think about the Gamma shifts of L-functions in general. (6) In a later episode, we will quote Freeman Dyson where he says that Yuri Manin hoped for a "quantization of mathematics". Actually, here is the entire quote: "Manin sees the future of mathematics as an exploration of metaphors that are already visible but not yet understood. The deepest such metaphor is the similarity in structure between number theory and physics. In both fields he sees tantalizing glimpses of parallel concepts, symmetries linking the continuous with the discrete. He looks forward to a unification which he calls the quantization of mathematics."
For those who're wondering why we're interested in the operations at 2:47, it's because under the Langlands correspondence, all L-functions are supposed to come from "automorphic representations". The above operations correspond to very natural ones on representations. Roughly speaking, addition corresponds to direct sum, multiplication is tensor product, Sym^2 (or Sym^n) takes the space of homogeneous polynomials in degree 2 (or n). If one believes in the correspondence, then these operations in the automorphic universe better correspond to operations in the L-function universe as well. This functoriality principle is still not totally understood. [ There's also the dual, which wasn't mentioned. ]
Great series 👍thankyou 🙂. It's interesting that addition of L-functions is defined as multiplication of the Dirichlet series. If an infinite dimensional vector space was formed using the integer powers of primes. Addition in the vector space would be multiplication in the projection into the rationals also.
That moment at 30:40 where the Prof. gives you that look, like: "Okay, we are finally getting to the point where we have to take the training wheels off, and I'm not sure you're ready." Priceless!
Is there a reason the LMFDB doesn't show the "sum of harmonics" for L-functions like you did in episode 1? Edit: The process that gives the prime counting function for the riemann zeta function, I mean
Good question. My guess would be (1) that there is an enormous (infinite) number of interesting things that one could choose to include on every LMFDB page. They have made some reasonable choices. (2) Even if you wanted to include something like the spikes we had in Episode 1 and 3, there are "better" (but less elementary) choices of "base function" than cos(t log(x)), giving rise to other graphs in the same spirit.
Another great video! In \mathbb{L}, when we are viewing it as being like a Hilbert space, Would it be as, “like a Hilbert space, except over the rig \mathbb(N} instead of over one of the fields \mathbb{R} or \mathbb{C}” ? Or, should there be a way to take like, L^{\oplus r} for rational numbers r? Seeing as you mentioned that the \otimes is conjectured but not proven to be total, but didn’t say something like that for \oplus , should I interpret this as implicitly indicating that \oplus has been proven to be total (for some axiomatization of what an L function is)? Hm, a thought: The “main steps” of these operations, these seem to be guaranteed to produce some kind of object, just, often not an L function, right? I imagine people have studied a fair bit already some questions about these missing-the-correction-to-make-an-L-function objects? Like, “if one computes the relevant series, does it still give a meromorphic function on all of \mathbb{C}?” (And, assuming it does) “do they completely fail to have the symmetry property that L functions have, or is is just kinda broken by something?”, “does the \otimes-without-correction-step still distribute over \oplus?”, etc. (idk whether these are the types of questions one would ask, just what seemed plausible to me) I suppose I should pick up one of those books, haha.
Yes, for \oplus it follows quite directly from the axioms in E4 that it is well-defined (or total). And regarding the scalars, you can choose which perspective you want to use \mathbb{N} is an option, \mathbb{Z} is how I would think of it, but you could always (technical term) "tensor with R or with C" to get other scalars beyond Z. There is nothing deep in the choice of scalars, just a convention you have to decide on. And of course, strictly speaking we shouldn't use the word "Hilbert space" unless we choose the scalars to be R or C. You are also totally right in that the "main step" of the product operation is something understood, but it's not at all easy to prove that the result will satisfy the axioms for an L-function.
@@PeakMathLandscape Ah! I should have thought of tensoring with R of C! That makes sense, thanks! (Though, I suppose in order to be sure that this wouldn’t lose any elements, you would have to show there’s no torsion? But I imagine that that’s not hard to show.)
I'm blown away. Just what I've been searching for. This has been a deep dive revelation for me. It touched on my wondering what several terms refereed to. Ramification, Grothendieck sheaves I suppose and the zeta function role in L functions as well as character coefficients. WOW!
I was trying to follow but at this point I got really lost. Is there any L-function encoding information about the precise divisors of a natural number? I mean, not only the sum or amount of divisors, but WHICH are those?
:-) We plan to put the release schedule up on the webpage (peakmath.org) but it's not quite ready for posting yet. Some time in September I think. Let us know if you find an L-function that does the job!
I don't think this would be a Hilbert space, but rather a Z-module, as I don't see any meaningful way to define non-integer multiples of L-functions without using non-integer powers of them, which would introduce branch cuts and all sorts of difficulties. And that's if we only make it a real Hilbert space!
I didn't get why addition combines like that at specific order? id doesnt look like regular distributive thing (x + y)(z + w) = xz + yz + xw + zw. Since they are infinite series how do you bound them to just few terms in kinda sporadic order?
This episode was the first one that flew straight past me. The operations fly in from nowhere without explanations, and I fail to see why you would use these things in the first place, like why are they necessary? "Trust me bro it'll come later" is not really satisfying, like at least some justification for what you're doing would be nice.
Not a full answer for sure, but many partial and potential future paths to the Riemann Hypothesis depend on these operations. One example: We cannot prove that all nontrivial zeroes are on the critical line, but we can prove that there is a "zero-free region" in the critical strip, so that is some partial progress towards GRH. The standard method employed is the method of Hadamard and de la Vallée Poussin, which relies on the two operations "addition" and "multiplication" here. This method works for rather general automorphic L-functions and is planned to appear in Episode 11. Also, it is hoped that a future proof of RH could be found through some new theoretical mechanism implying "positivity". The numbers that have to be proven positive here are precisely numbers you get by evaluating an additive invariant of L-functions. Additive invariants are a main theme of the next Episode, and we couldn't even define them without first talking about addition here.
yes...big like.....my '2^n^d' most favorite function the 'zeta function'... the zeta function connects prime point/points ..... to...... vacuum area patches of bounded prime numbers..... patches individually identify number fields Pn........as unique states of static (rest) matter 'm = Pn'.....with energy E = m * c * c points at orbits 'n'......next video maybe?
![[UNCUT] "บอสณวัฒน์" แฉ! แหล่งซุกเงินแก๊งบอส ขยันผิดที่ไม่กี่ปีก็เข้าคุก l คนดังนั่งเคลียร์ l17ต.ค.67](http://i.ytimg.com/vi/BL56qIfXYZM/mqdefault.jpg)
Curiously enough, I have been reading Fearless Symmetry as I'm following your wonderful series.
this whole world feels like relatively untapped potential for the youtube education space, almost surprised there seems to be so little digestible content about this stuff in this form, but i guess its because everything is so hand-wavy and conjectural at the moment. ive been loving the introduction nonetheless
Summation:
Being interested in Primes, and believe me - done a hell of a lot of research. TH-cam as a whole - only gives a basis.
These videos work though, and seriously condense everything currently available, all in nice little chunks.
Have learnt a massive amount more, and love the fantastic work / research / production / complexity involved.
Very very good!
Cannot wait for further episodes.
do you have any recommendations/resources for a high school student like me? any books/websites? i know some basic group/ring/field theory, btw
@@artemetra3262 , IMHO the most accessible intro into Modern Number Theory are the three volumes "Number Theory 1-3(Translations of Mathematical Monograph)" by Kato/Kurokawa/Saito, starting from basics, going to Class Field and Iwasawa theory to a sketch of Wiles' proof of Fermats Last Theorem. Lots of numerical examples. Highly recommended.
I feel like this episode is what finally tied everything all together for me and provided a good look into the math behind the magic of L functions. Maybe it's because I have a category theory background, but thjs just finally made it click
Me too, I believe this is the most important video introduction to Langlands Program I've ever seen.
I can't believe the channel has only 12.3K subscribes for such high quality content.
Was that an intentional reference to the Grothendieck prime?
Of course
I have a bachelor's degree in computer science, and I work as a computer programmer. But I've always had an interest in mathematics. I've been using SageMath for over 15 years to study things like tetration (iterated exponentiation, including an iteration count that is non-integer, perhaps even complex), the Riemann Hypothesis, and prime gaps (especially maximal prime gaps). But I have always been daunted by L-functions, so I've never explored beyond the Riemann zeta function. I have a fascination with the Riemann Hypothesis, but I have always felt too overwhelmed to look into the GRH.
This video series is opening new possibilities to me, and this episode especially has created a desire within me to study L-functions more thoroughly. Thank you for making these videos available to the public like this!
Thank you so much for making this series!!
When you said that maybe the Riemann Z function could perhaps be equivalent to the vacuum state, a shiver ran down my spine and goosebumps ran up and down my skin
You stole my comment, but I fully agree!
😎
I was unmoved, knowing he was bluffing, notice he gave no explanation. When mathematicians don't explain, he is bluffing.
Is the ⊕ operation on L-functions associative and commutative? Are there identity elements, and is the quasigroup property satisfied, or are there inverse elements? Or, are these currently all open questions in the research?
The plus operation (well done getting the actual symbol into a comment!) is well understood, and it has all the properties you could hope for (associative, commutative, identity element, inverse elements) although for the inverses you have to choose a convention where things like the Möbius mu function counts as an L-function. The Dirichlet series of the Möbius mu function is the inverse of Riemann zeta, and similarly, every L-function has it's own associated "Möbius-like" inverse. The identity element is the "trivial" L-function corresponding to the sequence 1, 0, 0, 0, 0, 0, ...
Amazing! My favorite content. I watch with big attention.
I was very surprised by the conjectural algebraic structure of L-functions, very interesting stuff
Also, loved Adelita by Tarrega at the end :)
Excellent video as usual
Eager to see where this takes us! I finished Love & Math and just ordered Fearless Symmetry.
Very good video with a great motivating intro. To strengthen the analogy with quantum physics, one would for example expect some "raising" and "lowering" operations to create states from the vacuum. Are there analogs to this here, maybe among the unary operations you presented? Indeed many things about L-functions remind me of quantum systems, for example the seemingly quantized "spectral parameters", i.e. the Gamma-shifts.
Viewers interested in quantum mechanics might want to check out my "Physics" playlist, where I aim to explain QM at a similar intermediate level of technicality, avoiding oversimplification.
Ah, deep question. There are general hopes for some form of "quantization" in the world of L-functions, but there are many different speculative ideas on how to make such a vague idea into rigorous mathematics. Impossible it is, to discuss all of this in TH-cam comments, but a few brief remarks.
(1) Google "Berry-Keating conjecture" or "Berry conjecture" for one direction.
(2) In a different direction, it feels like the real nontrivial zeroes behave like bosons, and the other nontrivial zeroes behave like fermions, in that the real zeroes can congregate into a multiple zero at the central point. Perhaps this is a nonsensical idea, perhaps it could lead somewhere. See Episodes 7 and 8 on the BSD conjecture!
(3) For raising and lowering operators, there are no obvious direct analogues as far as I know, but this is certainly an idea worth playing around with. If you start with a degree 2 L-function, say L_E, and apply the symmetric power operator Sym^k for k=1, 2, 3, 4, ... you get a sequence of L-functions of degrees 2, 3, 4, 5, ... and this is one of many possible starting points for playing around. Could you define a "raising operator" on L-functions that in some generality takes Sym^k L to Sym^{k+1} L???
(4) The reason for actually comparing Riemann zeta to the vacuum is that the Riemann zeta is conjecturally the only L-function with a pole at the edge of the critical strip. All other L-functions should be entire (i.e. no poles) except if they contain Riemann zeta as direct sum component. Could this pole be given a physical interpretation as "mass zero"???
(5) As you point out, the Gamma factor shifts look a bit quantized. For full appreciation of this, one should also look at transcendental L-functions, for example from Maass forms, where these Gamma shifts are not real, and seemingly transcendental, and where they appear with a close relationship to eigenvalues of a Laplacian operator. I don't think anyone has a complete understanding of how to think about the Gamma shifts of L-functions in general.
(6) In a later episode, we will quote Freeman Dyson where he says that Yuri Manin hoped for a "quantization of mathematics". Actually, here is the entire quote:
"Manin sees the future of mathematics as an exploration of metaphors that are already visible but not yet understood. The deepest such metaphor is the similarity in structure between number theory and physics. In both fields he sees tantalizing glimpses of parallel concepts, symmetries linking the continuous with the discrete. He looks forward to a unification which he calls the quantization of mathematics."
@@PeakMathLandscape Thank you for the many pointers.
For those who're wondering why we're interested in the operations at 2:47, it's because under the Langlands correspondence, all L-functions are supposed to come from "automorphic representations". The above operations correspond to very natural ones on representations. Roughly speaking, addition corresponds to direct sum, multiplication is tensor product, Sym^2 (or Sym^n) takes the space of homogeneous polynomials in degree 2 (or n). If one believes in the correspondence, then these operations in the automorphic universe better correspond to operations in the L-function universe as well. This functoriality principle is still not totally understood.
[ There's also the dual, which wasn't mentioned. ]
I am honestly excited for each new episode
I like this season looking forward for new videos
Great series 👍thankyou 🙂. It's interesting that addition of L-functions is defined as multiplication of the Dirichlet series. If an infinite dimensional vector space was formed using the integer powers of primes. Addition in the vector space would be multiplication in the projection into the rationals also.
At first I'm wondering where this is all going, but the ending showing L-functions as equivalent to states in Quantum mechanics is pretty insane.
That moment at 30:40 where the Prof. gives you that look, like: "Okay, we are finally getting to the point where we have to take the training wheels off, and I'm not sure you're ready." Priceless!
excellent video as always. Keep up the good work❤
How I wish I had had teachers like you
How long until the next of these? I'm addicted!
Around 3 minutes now.
Is there a reason the LMFDB doesn't show the "sum of harmonics" for L-functions like you did in episode 1? Edit: The process that gives the prime counting function for the riemann zeta function, I mean
Good question. My guess would be (1) that there is an enormous (infinite) number of interesting things that one could choose to include on every LMFDB page. They have made some reasonable choices. (2) Even if you wanted to include something like the spikes we had in Episode 1 and 3, there are "better" (but less elementary) choices of "base function" than cos(t log(x)), giving rise to other graphs in the same spirit.
There exists an infinite number of L - functions.
La being the "tensor square root" of identity, Lp, seems to be an interesting feature.
Yes! But there are many (in fact infinitely many) tensor square roots of the identity, and L_A is just one of these.
Another great video!
In \mathbb{L}, when we are viewing it as being like a Hilbert space,
Would it be as, “like a Hilbert space, except over the rig \mathbb(N} instead of over one of the fields \mathbb{R} or \mathbb{C}” ?
Or, should there be a way to take like, L^{\oplus r} for rational numbers r?
Seeing as you mentioned that the \otimes is conjectured but not proven to be total, but didn’t say something like that for \oplus , should I interpret this as implicitly indicating that \oplus has been proven to be total (for some axiomatization of what an L function is)?
Hm, a thought: The “main steps” of these operations,
these seem to be guaranteed to produce some kind of object, just, often not an L function, right?
I imagine people have studied a fair bit already some questions about these missing-the-correction-to-make-an-L-function objects? Like, “if one computes the relevant series, does it still give a meromorphic function on all of \mathbb{C}?” (And, assuming it does) “do they completely fail to have the symmetry property that L functions have, or is is just kinda broken by something?”, “does the \otimes-without-correction-step still distribute over \oplus?”, etc. (idk whether these are the types of questions one would ask, just what seemed plausible to me)
I suppose I should pick up one of those books, haha.
Yes, for \oplus it follows quite directly from the axioms in E4 that it is well-defined (or total).
And regarding the scalars, you can choose which perspective you want to use \mathbb{N} is an option, \mathbb{Z} is how I would think of it, but you could always (technical term) "tensor with R or with C" to get other scalars beyond Z. There is nothing deep in the choice of scalars, just a convention you have to decide on. And of course, strictly speaking we shouldn't use the word "Hilbert space" unless we choose the scalars to be R or C.
You are also totally right in that the "main step" of the product operation is something understood, but it's not at all easy to prove that the result will satisfy the axioms for an L-function.
@@PeakMathLandscape Ah! I should have thought of tensoring with R of C! That makes sense, thanks!
(Though, I suppose in order to be sure that this wouldn’t lose any elements, you would have to show there’s no torsion? But I imagine that that’s not hard to show.)
@@drdca8263 Exactly, no torsion here
I'm blown away. Just what I've been searching for. This has been a deep dive revelation for me. It touched on my wondering what several terms refereed to. Ramification, Grothendieck sheaves I suppose and the zeta function role in L functions as well as character coefficients. WOW!
These are the elements of a great THEORY.
I was trying to follow but at this point I got really lost.
Is there any L-function encoding information about the precise divisors of a natural number? I mean, not only the sum or amount of divisors, but WHICH are those?
Is there an L-function, nontrivial zeroes of which gives gives us the video release schedule if we multiplied them by logx and put into cos?
:-) We plan to put the release schedule up on the webpage (peakmath.org) but it's not quite ready for posting yet. Some time in September I think. Let us know if you find an L-function that does the job!
Wonder how deep will be going with grothendiec stuff.
Honestly, up to this point many things are still incredibly random to me! Will this ever stop?
is the size of L is countable?
Muy interesante. 👏👍
For a moment I thought the allegorical introduction was about the whole Langlands program...
PS: But then again, it's all related!
Hi!
How can i as a just new University math student follow this series? Is it possible or should i wait to try untill i get my degree?
Only you can judge for yourself! Try it and see whether you find it meaningful or not.
I don't think this would be a Hilbert space, but rather a Z-module, as I don't see any meaningful way to define non-integer multiples of L-functions without using non-integer powers of them, which would introduce branch cuts and all sorts of difficulties. And that's if we only make it a real Hilbert space!
I didn't get why addition combines like that at specific order? id doesnt look like regular distributive thing (x + y)(z + w) = xz + yz + xw + zw.
Since they are infinite series how do you bound them to just few terms in kinda sporadic order?
I'm not going back to crossword puzzles.
This whole series has kind of taken the mystery of advanced mathematics away.. it does not seem all that exciting anymore :(
This episode was the first one that flew straight past me. The operations fly in from nowhere without explanations, and I fail to see why you would use these things in the first place, like why are they necessary? "Trust me bro it'll come later" is not really satisfying, like at least some justification for what you're doing would be nice.
Not a full answer for sure, but many partial and potential future paths to the Riemann Hypothesis depend on these operations.
One example: We cannot prove that all nontrivial zeroes are on the critical line, but we can prove that there is a "zero-free region" in the critical strip, so that is some partial progress towards GRH. The standard method employed is the method of Hadamard and de la Vallée Poussin, which relies on the two operations "addition" and "multiplication" here. This method works for rather general automorphic L-functions and is planned to appear in Episode 11.
Also, it is hoped that a future proof of RH could be found through some new theoretical mechanism implying "positivity". The numbers that have to be proven positive here are precisely numbers you get by evaluating an additive invariant of L-functions. Additive invariants are a main theme of the next Episode, and we couldn't even define them without first talking about addition here.
yes...big like.....my '2^n^d' most favorite function the 'zeta function'... the zeta function connects prime point/points ..... to...... vacuum area patches of bounded prime numbers..... patches individually identify number fields Pn........as unique states of static (rest) matter 'm = Pn'.....with energy E = m * c * c points at orbits 'n'......next video maybe?
Why do you think he says serious instead of series. He is a Martian.
LOL
Which part? The analogy about “what if there was a world where people just studied integers individually and not operations on them?”
What is there to study about number 5? This video makes no sense
It's very advanced.
This whole series means nothing to me
It's very advanced. Begin with complex variables or calculus and various Algebra first.
L functions are in the complex plane or the upper half plane? Seems both views are possible.