a bit about one of Ramanujan's favorite functions
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- Опубліковано 23 кві 2024
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Me: Why 24?
Michael: Because 24=26-2
Me: Damn! That's unbelievable!
I had the exact same thought. Michael: “Why 24? 24 is 26 minus 2, and 26 is a critical dimension in super string theory.” That has to be the most nonsensical thing I’ve ever heard in one of his videos.
it's a baffling statement without further explanation but afaik it's actually a perfectly sensible thing to say with some background knowledge. I don't really understand any of it, mind you, but for some reason that 26D theory involves a 24D symmetry group, the Leech Lattice. I think it's something like, you have a space of 24D points but in string theory you are talking about 1D objects, wiggly curves, basically, so that adds one dimension, and then since you have time (the strings evolve in time), you get another, landing you at 26? But please don't quote me on that
Also related somehow is the fact that
Sum_(n=0)^24 n² = 4900 = 70²
i.e. the sum of the first 24 squares is again a square, which, afaik, is a very unique property. It kinda means that, if you take one additional step in every orthogonal direction through 24D Euclidean space, taking 300 steps total, you end up exactly 70 steps from where you started.
24 is also 42 in reverse so (by Michael's reasoning) this shows a deep connection between number theory and the Hitchhiker's Guide to the Galaxy!
6:26 michael penn is a vampire confirmed
😮
"1472 is my birth year" is the best bait for increasing the number of comments under the video. Well done, 552-year-old man
He doesn't look a day over 550.
0 is really important because it's 26 less than 26 which is an important number in superstring theory
24 is important because it is 4!, and also the largest factorial that doesn't have 0 as its last digit.
Correction: 26 is the critical dimension in bosonic string theory. The critical dimension in superstring theory is 10.
There should be a follow up video to explain this video...
I’ve always managed to understand all of your videos. I did understand literally nothing of this one.
Not saying this is the key, but it helped me to write out the sum and product a bit.
Same with me! Why the hell is tau(1)=1 and tau(2)=-24 ?????
@@wolliwolfsen291because if you expand the product you will see (as he showed) that the coefficient of q¹ is 1 and the coefficient of q² is -24
"Today" made sense to me.
@@lorenzosaudito But what do those coefficients have to do with the values of the function?
6:25 - My man was born in the XV century.
Of course, as many will no doubt recall, 24 (or it's reciprocal) appears twice in Ramanujan's most famous, and most incredible Partition Function p(n) (or the asymptotic expansion for it), which is a wonder to behold! What a Guy!
Today's date is also 24 th April
24 is the number of sulutions to x^2+y^2+z^2+w^2=1 in the F4 lattice, also 24d hosts the leech lattice
If you reverse the digits of 24 you get 42.
You said (1-q^m)^7 is congruent to 1-q^(7m) because the factors of the rest of the terms are multiples of 7.
But q itself is not an integer, the series doesn't even converge beyond |q|
In this context, q is considered a "formal variable" or the series as a whole to be a "formal power series", which means we're not concerned with questions of convergence and all we're using q for is as a variable to keep track of the coefficients that we're interested in.
There are "q-analogs" of many things, for example of integers [n]_q = 1 + q + q^2 + ... q^(n-1), and q-binomial coefficients can be defined by first defining q-factorials [n!]_q = [n]_q[n-1]_q...[1]_q. Then the q-binomial coefficient can be defined analogous to the usual way, [n choose m]_q = [n]_q/([m]_q[n-m]_q).
That's not to say the other perspective of actually evaluating the expression isn't useful - it really is! You may notice that [n]_q at q=1 is equal to n, and similarly with n! and (n choose m).
But what about other values of q - what would it mean to plug in q=-1? Or even complex numbers like q = i? Well this area of study (at least in the combinatorics context I'm familiar with) is called "cyclic sieving". And it turns that substituting q=-1 often tells us something about reflexive symmetry, and in some cases, plugging in a kth root of unity will tell us things about the objects the generating function was defined from, under some cyclic permutation of the objects of order k.
Here's an explicit example. Let S(4,2) be the set of 2-element subsets of {1,2,3,4}, which are {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}.
The q-binomial coefficient [4 choose 2]_q = (1+q+q^2+q^3)(1+q+q^2)/(1+q) = (1+q^2)(1+q+q^2) = 1 + q+2q^2+q^3+q^4.
Now consider the symmetry of adding +m taken mod 4. The imaginary number i has cyclic symmetry of order 4, since i^4 = 1, so this is what we'll use for q. If we add 0, nothing changes, so q = i^0 = 1 tells us [4 choose 2]_q = 6, and we have 6 fixed points.
If we add 1, then we get {1,2} -> {2,3} -> {3,4} -> {1,4} and {1,3} {2,4}. We have no fixed points, and we see at q = i^1 = i, we have 1 + i + 2i^2 + i^3 + i^4 = 1 + i - 2 - i + 1 = 0.
If we add 2, then now we see {1,2} {3,4} and {2,3} {1,4} but we have 2 fixed points {1,3} and {2,4}, and in my opinion quite amazingly, we see [4 choose 2]_q at q = i^2 = -1 is equal to 1 - 1 + 2 - 1 + 1 = 2.
I love your channel!
Are you going to do nxet something with modular forms? I want to understand their beauty.
Literally this very video is about modular forms in disguise lol.
@@sbares ..and exactly therefore I'd like a dedicated video for that
11:53 The identity on the left of the board says that the sum equals an infinite product of 1-q^m factors over all m. But you seem to substitute it here just for a single factor from that product. Did I miss something?
He has only one (1-q^m)^3 factor on the left hand side, he then needs the product of all, which is exactly what is needed for the identity. Michael puts that sum one step before where is indeed needed.
The video is quite chaotic...
Yes, and the sum does not need to be cubed, according to the Jacobi Identity.
I'm confused by the use of the Jacobi Identity... in the definition on the left the sum is not qubed but when you use it the sum suddenly is... I have a feeling that googling "Jacobi identity" isn't going to help because theres at least one that's by far more known (Lie algebra)
There's Jacobi's triple product but it looks different from this identity.
There's a proof of Jacobi's triple product in Apostol's Introduction to Analytic Number Theory, Section 14.8 and more specifically in connection with the Ramanujan Tau function in Bump's Automorphic forms and representations, Section 1.3 and the exercises therein.
@@user-oe5eg5qx4c The identity follows from the triple product. It's theorem 1.3.9 in Berndt's book.
I think it’s 24 because π/12 is a nice and natural way to divide a unit circle.
Why is q used in these functions and not x? Is there a meaning behind it?
In number theory the q is used and is usually more like a formal series without taking care of convergence.
11:40 why is the summation cubed?
Multiplying q into the sum later on looks wrong too (11:40)
I don't know why the summation is cubed, but it shouldn't be, according to the Jacobi Identity.
1472 bruhh
24 is a very important number as it has two of the digits of Michael Penn's birth years in reverse order which is also orthogonal to my understanding of this video
Well, that was exactly the 300 million different birth year that Michael has given
I had to prove once that if a Dirichlet character only has real outputs, then its group that it's defined over is (Z/mZ)*, where m divides 24. I vaguely remember that it had something to do with the fact that 24 is one less than 25, which is 5^2. There was something special about the prime 5, and square came from the fact that elements needed to have order 2. I don't recall all the details, but 24 is indeed a bit special when it comes to number theory.
A little poking around seems to show that the 24 comes from the Leech lattice, which doesn't exactly help me either, but at least it's something.
Always loved the birth year thing
On a limb here, waning in the wind, on a waning moon, ...would say that;
24 is the ordinal complexity of a tautology for the axiom of choice, 23 being of soundness, 22 of completeness, and 21 of compactness, the 21 best verified by using a derivative 5 times, and thus justifying categoricalicity w/o compactness?
Looks like his L-function puts categoricalicity in stone?
12:40 The way you have written that here, shouldn't that be a q^(1/3 + n(n+1)/2) ? Or whas the ³ supposed to include the q?
Ah! Was this on the exam or final or quiz?? Or chapter 11
I have a PhD in algebraic number theory, and I have never seen this function before. I have to ask, what's it's purpose?
i was thinking the same thing: "really important function in number theory" - because it's a function that is divisible by 7 when you input multiples of 7. guess what: the identity function does that too xD
"What's that 24 doing there, Ramanujan?"
"It was revealed to me by the gods"
12:08 The "cube" of the last expression isn't needed because the LFS of Jacobi Identity already has a "cube."
Or the that of "Jacobi Identity" isn't needed?
It's different from ua-cam.com/video/aZiCcO8H4jI/v-deo.html
@8:47
“1472; that’s my birth year”
Umm, Michael, I somehow don’t believe you.
When you brought the q into the cubed sum, it should have been raised to 1/3 power...
I love all your others videos, but bro what in the world is happening rn (I did not understand what was going on during the mod 7 thing can someone explain). Also, wdym ur born in 1472?
I really like number theory, but I get lost easily, not my strong point.
hate to ask - but what's the pt. of this ?
552 years old and counting
1472
1/4 of 72 = ⌊video length⌋
:|
You needed to explain the start better, i was trying to wrap my head around some kind of converging function instead of just thinking of an infinite binomial series.
You still lost be at the Jacobi stuff, but that was my fault
It actually does converge (for |q| < 1, or equivalently for q = exp(2*pi*i*z) with z in the upper half-plane) to a function known as the modular discriminant.
More importantly, 24 is almost as funny as 25.
You can't bring q inside the cubed sum at 12:57. The correct equation is q(sum ...)^3 = q^3/q^2 (sum ...)^3 = q^(-2) (sum ... * q)^3.
I have the same question. q^{-2} seems to be lost.
I figured out. The sum should NOT be cubed. It is a typo.
@@alexeycanopus1707 So where does the q go? Is Jacobi's identity not written properly?
@@aadfg0 jacobi’s identity should not be cubed.
@@aadfg0 I lost “NOT” where I wrote a comment. I apologize
Bruce Berndt.
Vote for Pedro!
these surprising connections between math and quantum reminded me of this video: d1IzNKIHhp0
You know what number is funnier than 24? 25
:)
Your videos are usually very good, but this one was disorganized and impossible to follow. And your statement that the expression is raised to the 24th power because “24 is 26 minus 2, and 26 is an important dimension in super string theory” couldn’t possibly make less sense.
2nd!
Understood almost nothing at all
I understood nothing about the video.
24 is 26 -2 so it connects...... ah ah ah ah ah never heard a more stupid proposition. The true is 24 read from right to left is 42 the meaning of life. ah ah ahah aha.
I mean, that's literally the connection. Of course, he didn't explain the details so it sounds like magic, but it is literally the reason the critical dimension is 26.