For my degree dissertation I wrote about Analytic Number Theory. I dedicated a whole chapter to Moebius function. I never got a deep intuition for the function, but this video changed that! Thank you
I can't watch it entirely now but I don't want the algorithm to punish the video. I'll keep going later. I didn't know the SOME 4 was already on going. Great news!
This video is incredibly good. It is presented in a way that made me want to make notes, and make notes i did! Very well structured as well, laying down the relevant foundations needed to understand the next part, and the next, and so forth. I'm definitely going to watch more of these videos if they encourage me to write down notes like this one.
Also, we can use the Euler product of the zeta function: ζ(s) = prod (1 + p^-s + p^-2s + ...) = prod (1 - p^-s)^-1 so 1/ζ(s) = prod (1 - p^-s), and expanding it out, we see the coefficient on n^-s matches the definition for μ(n)
Sir, that is a great video! Its been a while since i watched a math- related video and made me want to explore a part of math that i havent discovered yet. Number theory seems like a really interesting concept, and i say that only based on how you present it on this video, so i thank you and congratulate you for making such excellent content
Here's another way to look at the mobius function from number theory. Similar to how we constructed the mobius function as the inverse of the dirichlet series having all ones, we can do the same thing with the power series having all ones, i.e. 1 + x + x^2 + ... = 1/(1-x) . The inverse of this is 1-x, the mobius function for the case of power series. One can think of mobius functions more generally for posets, and the one for power series is the simplest one, it is the infinite poset given by the natural numbers under the
Wow, great video! I felt like as the video kept going it just gotten better and better (and I really liked the visualisation at the end) Btw I think a video that shows some possible usages of these concepts could be really cool
Same thing. A sequence is really just a function from the natural numbers to something else. So you can write it with an index or as a kind of function call, it's the same thing.
@@AllAnglesMath sorry I am confused. I though that a,b is a wired polynominal where instead od x^n it has n^x. So a(x) should calculate polynominal value for x. But if a,b is a sequence its elements should be separated by , not +. All this does not change that this is excelent video, I am just cinfused with syntax used.
@@adrian_sp6def Ah, now I understand. You're absolutely right that my notation is confusing. I am using the same letter 'a' for both the "weird polynomial" and also for its sequence of coefficients. I should have used different letters to make the distinction more explicit. Very sorry. So, to be absolutely clear: In the sum at the bottom, a(p) refers to the p'th coefficient a_p of the weird polynomial, not to the evaluation of the entire polynomial. My bad. Thank you so much for pointing this out.
💛 I may add to the final note: although a definition kinda presents itself, there are infinitely many possible constructions in maths that can be done, but only some of them get selected because they’re cleaner to work with (and for example define!) and they are more connected to other things (and often applications). From this standpoint, one can ask why had μ as it is made it into mainstream theory, if it has a definition that seems a bit clunky, but you show how it’s also simply related to the 1̅ sequence and also it can be noted from Hasse diagrams that in the end the definition itself is quite graspable too: alternating ±1 parity thing happens very often in other math, and adding a 0 clause is only too useful if we want to deal with square-nonfree numbers at all (and we do!). Videos that touch this process of working with constructions-and even more, choosing one over another (which is undoubtedly harder to show, so I’m okay with that being presented rarer in popularization/explainers-I get how much work it is to make a good clean thing)-are so important to make people consider trying doing more math of their own and experimenting and not just staring at some text without an idea that it can be visualized and can be understood by themselves. Thanks!
Thank you for the kind comment. We have made another video that follows the same idea: we show that there is only a single correct formula for 2D multiplication that satisfies all the required properties. It turns out to be the complex multiplication. You can find it here: ruclips.net/video/di5QKO9xg2I/видео.html It's often very instructive to show why a definition is "inevitable" once you have agreed on its important properties.
Just a reminder: SoME4 will, presumably, take place in 2025. This is "only" SoMEπ. It is community-driven and there will be no price money and no 3Blue1Brown top 5 video this year.
I do a lot of reading, watching other people's videos, ordering books, going through articles, ... Sooner or later you run into interesting stuff. e.g. I found Hehner's approach in the Mathematical Intelligencer journal at the library where I used to work.
Sometimes things do not "slowly evolve" but just pop out when doing something. Sometimes one makes some definition because it is helpful and later on finds out where it fits. E.g., |μ| is something that someone would easily come up with as an "indicator" function for squares because indicator functions are very useful as they basically represent the idea of "has property" functionally which means they can be used as functions in calculus rather than trying to do everything as sets. This then lets one use the theory of functions to study things that may not really be that obvious as sets. It's obviously true then that |μ| = sgn(μ)μ which is equivalent to μ = sign(μ)|μ| then the question is "what could the sgn(μ) be"? While in some cases there might be many choices it is obvious that in this case a natural choice is quite limited since we are talking about the prime factorization and there is a limited number of choices with square free: p1^a1*p2^a2*...*pn^an. ak < 2 else the integer is not square free. So all we have in square free integers are a product of "singular primes". So all we can really do is count them(that is the most natural thing) and talk about the parity of the number. Hence sgn(μ) = (-1)^(# of prime factors in factorization of a square free integer). So the point here is that coming up with μ is really not that difficult and something likely that didn't take that long and many people came up with it or the variant |μ| and this likely happened even thousands of years ago when people were thinking of prime numbers and thinking about just multiplying singular primes(which then they are ultimately thinking about |μ|) and maybe wanting to distinguish between having even and odd number(which gives μ). What took time is to develop and see how μ showed up in a variety of places and how it could be used as a fundamental function in number theory. There are likely millions of functions people have defined easily/quickly using similar logic that have not yet been shown to be connected to any deep theory. Usually the simpler the function the more places it will show up. μ is quite simple. It's about as simple as one can get when talking about primes and trying to investigate their exponents. Primoridal primes[square free integers] come up with often when looking at integers because they sort of act as the prime building blocks "2nd stage". E.g., if one is doing sieves and such. Because any integer is either a Primoridal or a product of Primoridals. So I would say, at least in this case, that μ is not complex at all. It only seems that way at first because number theory is pretty unnatural for people. Almost no one things to investigate numbers in the way that number theory does. But once you realize numbers have all these intrinsic relationships and start thinking about them and how they interact you'll very likely to stumble upon |μ| sooner or later. In your video, as you explain, when you move multiplying and involving polynomials to that of Dirichlet series(which isn't a leap) then one naturally will seek out μ. These things can happen out of the blue when one studies such things. Of course we have hindsight but I think most people that have spent years doing math everyday can attest to how they figure out "new" things only to learn they already existed. It's sorta like once you get going down a road you're gonna start coming across the same things other people have went down that road before have saw. If you are the first to go down that road it might be a little slower and you are the first to see those things but sometimes it just all works out surprisingly well. Basically almost anything is going to be "foreign" to someone that doesn't spend time learning about something. The more time one spends on something the more "patterns" their brain will automatically come up with. Probably the biggest issue with people on the forefront of knowledge is that they are traveling down a dark road and don't know if it will yield anything of value. Usually the people that care less(and can care less) about where the road is taking them and only care about the sites they see are the ones that make the most progress going down the road(which is never ending).
Great video. But I couldn't understand after around 3:40. I've heard about a related concept, Möbius transformations from a popular book (VCA) by Tristan Needham. I still don't really see the connection yet other than they are both connected to group theory. Perhaps I need to rewatch the video a few times and reread VCA a few times to understand it. I'm glad that some4 is happening, and not some3.5 as I've previously heard.
While they're named after the same person, the moebius function here is otherwise *completely unrelated* to moebius transformations as you would see in Complex Analysis. So it's not weird that you don't see a connection: there isn't one.
Basically what @diribigal said. That Möbius guy must have been very busy, because he has his name on multiple unrelated concepts. By the way: I love VCA, it's an amazing book. But, indeed, unrelated to this video.
I've written my own animation library over the years. If you want to program your animations, your best option is probably Manim ( www.manim.community/ ). But you can also make animations by hand, with simple tools such as powerpoint. Good luck with your video!
Grant cancelled #4 on October 16th, 2023. He wants to do it every other year. Too much work for him, I guess. Thanks for the video though. Expect me in your comment section asking dumb questions.
Hey, numberphile discussed this once Edit: in the form of Merten's conjecture, which is about whether or not the sum of the moebius function across the first n numbers is bounded by √n
Not all functions are formulated from a dirichlet series. Dirichlet series and functions are also two different things but if you regard the dirichlet series as a function, then they only classify as a small subset of all functions as they follow a specific form as seen in the video. If you are given a dirichlet series and want to see how it s composed, however, that’s a different story
Wonderfully explained. Thanks. It’s strange to me how so many maths textbooks just hand you definitions without really motivating them. It’s even stranger to me that many people don’t seem to have a problem learning that way.
It is not the inverse of the neutral element 1,0,0,0... is the neutral element 1,1,1,1... is the element taken in consideration The function of the video is the inverse of the 1,1,1,1... sequence
Yeah! Anyway, anytime you find that a neutral element has a different inverse from itself, it signals that either it wasn’t the neutral element, or it’s not the operation under which it’s neutral (compared to the operation wrt its inverse is a different thing), or maybe even they are indeed the same modulo some natural equivalence we forgot about, or something like that. And small errors like those do happen all the time, so it shouldn’t be any worry-despite math allows exact proofs of things by its very nature and it isn’t technically a natural science where you _absolutely need_ more varied evidence from the universe to be more certain about your theories, math still enjoys more evidence for one to become more certain in precisely the same way, so checking things from different angles (or considering particular cases and generalizations) is always a good idea even if you’re sure you have everything already proved by that point. And moreso if you’re not certain because the more insight and intuition one has, the better the road, and the more abundant are opportunities.
![Felix "Unfair" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/Oswujxm2Ag0/mqdefault.jpg)
Moebius function is my favorite function of all time. i love the part when it starts Moebing on all the numbers. truly a 10/10 function
Truly one of the functions of all time
noooooooooo
The RUclips comments just can't escape this meme. Just take my thumbs up.
Lmao
For my degree dissertation I wrote about Analytic Number Theory. I dedicated a whole chapter to Moebius function. I never got a deep intuition for the function, but this video changed that! Thank you
I can't watch it entirely now but I don't want the algorithm to punish the video. I'll keep going later. I didn't know the SOME 4 was already on going. Great news!
the 3b1b team doesnt have enough time and/or people to officially host SoMe4, so it's community-organized this year
@@wyboo2019 Ooh, interesting !
Brilliant, one of the best entries I've seen after all these years.
This video is incredibly good. It is presented in a way that made me want to make notes, and make notes i did!
Very well structured as well, laying down the relevant foundations needed to understand the next part, and the next, and so forth.
I'm definitely going to watch more of these videos if they encourage me to write down notes like this one.
Thanks for the positive comment. I hope you will learn a lot and ask intriguing questions.
goldmine, have been trying to understand the mobius function for the last few days, absolutely beautiful explanation, thanks
Happy to bring some clarity.
I loved the generatingfunctionology chapter on this. I actually love that book
Yeah, that's an amazing book. We should definitely do some videos about it.
I prefer the book by Flajolet and Sedgewick, it's incredibly extensive yet it's still very good on a pedagogical level
@@AllAnglesMath Yes, please! :) Also, excellent video :)
Also, we can use the Euler product of the zeta function:
ζ(s) = prod (1 + p^-s + p^-2s + ...) = prod (1 - p^-s)^-1
so 1/ζ(s) = prod (1 - p^-s), and expanding it out, we see the coefficient on n^-s matches the definition for μ(n)
Thanks
Excellent video! I really liked your approach, simple and straight forward. I hope to see more from you in this area.
Sir, that is a great video!
Its been a while since i watched a math- related video and made me want to explore a part of math that i havent discovered yet. Number theory seems like a really interesting concept, and i say that only based on how you present it on this video, so i thank you and congratulate you for making such excellent content
Here's another way to look at the mobius function from number theory.
Similar to how we constructed the mobius function as the inverse of the dirichlet series having all ones, we can do the same thing with the power series having all ones, i.e. 1 + x + x^2 + ... = 1/(1-x) . The inverse of this is 1-x, the mobius function for the case of power series. One can think of mobius functions more generally for posets, and the one for power series is the simplest one, it is the infinite poset given by the natural numbers under the
That is so cool! Thanks for sharing.
The best video of number theory I've seen! Thanks!
Thank you!
Wow, great video!
I felt like as the video kept going it just gotten better and better (and I really liked the visualisation at the end)
Btw I think a video that shows some possible usages of these concepts could be really cool
Just wow. Amazing. Thanks again
Quedtion for 5:40
The convolution formula has sum over a(p)b(q) is really an evaluation of a(x) or just an a index x for example a_1 or a_2?
Same thing. A sequence is really just a function from the natural numbers to something else. So you can write it with an index or as a kind of function call, it's the same thing.
@@AllAnglesMath sorry I am confused. I though that a,b is a wired polynominal where instead od x^n it has n^x. So a(x) should calculate polynominal value for x. But if a,b is a sequence its elements should be separated by , not +. All this does not change that this is excelent video, I am just cinfused with syntax used.
@@adrian_sp6def Ah, now I understand. You're absolutely right that my notation is confusing. I am using the same letter 'a' for both the "weird polynomial" and also for its sequence of coefficients. I should have used different letters to make the distinction more explicit. Very sorry.
So, to be absolutely clear: In the sum at the bottom, a(p) refers to the p'th coefficient a_p of the weird polynomial, not to the evaluation of the entire polynomial.
My bad. Thank you so much for pointing this out.
💛
I may add to the final note: although a definition kinda presents itself, there are infinitely many possible constructions in maths that can be done, but only some of them get selected because they’re cleaner to work with (and for example define!) and they are more connected to other things (and often applications). From this standpoint, one can ask why had μ as it is made it into mainstream theory, if it has a definition that seems a bit clunky, but you show how it’s also simply related to the 1̅ sequence and also it can be noted from Hasse diagrams that in the end the definition itself is quite graspable too: alternating ±1 parity thing happens very often in other math, and adding a 0 clause is only too useful if we want to deal with square-nonfree numbers at all (and we do!).
Videos that touch this process of working with constructions-and even more, choosing one over another (which is undoubtedly harder to show, so I’m okay with that being presented rarer in popularization/explainers-I get how much work it is to make a good clean thing)-are so important to make people consider trying doing more math of their own and experimenting and not just staring at some text without an idea that it can be visualized and can be understood by themselves. Thanks!
Thank you for the kind comment. We have made another video that follows the same idea: we show that there is only a single correct formula for 2D multiplication that satisfies all the required properties. It turns out to be the complex multiplication. You can find it here: ruclips.net/video/di5QKO9xg2I/видео.html
It's often very instructive to show why a definition is "inevitable" once you have agreed on its important properties.
@@AllAnglesMath 👍👍👍🤩
Just a reminder: SoME4 will, presumably, take place in 2025. This is "only" SoMEπ. It is community-driven and there will be no price money and no 3Blue1Brown top 5 video this year.
Loved the video! How do you find those specific/niche subjects ? (I'm also thinking about the video on unified algebra for example)
I do a lot of reading, watching other people's videos, ordering books, going through articles, ... Sooner or later you run into interesting stuff. e.g. I found Hehner's approach in the Mathematical Intelligencer journal at the library where I used to work.
It’s moebin time
Watching this and understanding everything makes me hope to fix at some point my relationships with number theory)
really really good candidate for some4
Sometimes things do not "slowly evolve" but just pop out when doing something. Sometimes one makes some definition because it is helpful and later on finds out where it fits.
E.g., |μ| is something that someone would easily come up with as an "indicator" function for squares because indicator functions are very useful as they basically represent the idea of "has property" functionally which means they can be used as functions in calculus rather than trying to do everything as sets. This then lets one use the theory of functions to study things that may not really be that obvious as sets.
It's obviously true then that |μ| = sgn(μ)μ which is equivalent to μ = sign(μ)|μ| then the question is "what could the sgn(μ) be"? While in some cases there might be many choices it is obvious that in this case a natural choice is quite limited since we are talking about the prime factorization and there is a limited number of choices with square free: p1^a1*p2^a2*...*pn^an. ak < 2 else the integer is not square free. So all we have in square free integers are a product of "singular primes". So all we can really do is count them(that is the most natural thing) and talk about the parity of the number. Hence sgn(μ) = (-1)^(# of prime factors in factorization of a square free integer).
So the point here is that coming up with μ is really not that difficult and something likely that didn't take that long and many people came up with it or the variant |μ| and this likely happened even thousands of years ago when people were thinking of prime numbers and thinking about just multiplying singular primes(which then they are ultimately thinking about |μ|) and maybe wanting to distinguish between having even and odd number(which gives μ).
What took time is to develop and see how μ showed up in a variety of places and how it could be used as a fundamental function in number theory. There are likely millions of functions people have defined easily/quickly using similar logic that have not yet been shown to be connected to any deep theory. Usually the simpler the function the more places it will show up. μ is quite simple. It's about as simple as one can get when talking about primes and trying to investigate their exponents. Primoridal primes[square free integers] come up with often when looking at integers because they sort of act as the prime building blocks "2nd stage". E.g., if one is doing sieves and such. Because any integer is either a Primoridal or a product of Primoridals.
So I would say, at least in this case, that μ is not complex at all. It only seems that way at first because number theory is pretty unnatural for people. Almost no one things to investigate numbers in the way that number theory does. But once you realize numbers have all these intrinsic relationships and start thinking about them and how they interact you'll very likely to stumble upon |μ| sooner or later. In your video, as you explain, when you move multiplying and involving polynomials to that of Dirichlet series(which isn't a leap) then one naturally will seek out μ. These things can happen out of the blue when one studies such things. Of course we have hindsight but I think most people that have spent years doing math everyday can attest to how they figure out "new" things only to learn they already existed. It's sorta like once you get going down a road you're gonna start coming across the same things other people have went down that road before have saw. If you are the first to go down that road it might be a little slower and you are the first to see those things but sometimes it just all works out surprisingly well.
Basically almost anything is going to be "foreign" to someone that doesn't spend time learning about something. The more time one spends on something the more "patterns" their brain will automatically come up with. Probably the biggest issue with people on the forefront of knowledge is that they are traveling down a dark road and don't know if it will yield anything of value. Usually the people that care less(and can care less) about where the road is taking them and only care about the sites they see are the ones that make the most progress going down the road(which is never ending).
You make some excellent points. The idea was just to give people some confidence when confronted with exotic definitions.
Great video. But I couldn't understand after around 3:40. I've heard about a related concept, Möbius transformations from a popular book (VCA) by Tristan Needham. I still don't really see the connection yet other than they are both connected to group theory. Perhaps I need to rewatch the video a few times and reread VCA a few times to understand it. I'm glad that some4 is happening, and not some3.5 as I've previously heard.
While they're named after the same person, the moebius function here is otherwise *completely unrelated* to moebius transformations as you would see in Complex Analysis. So it's not weird that you don't see a connection: there isn't one.
Basically what @diribigal said. That Möbius guy must have been very busy, because he has his name on multiple unrelated concepts. By the way: I love VCA, it's an amazing book. But, indeed, unrelated to this video.
and both are (at least probably) entirely unrelated to the single-faced Möbius strip
What an awesome video.
Very cool! Thank you.
I think we found the winner.
Great video.
Hey, I was just wondering, what did you use to make your videos? I want to do my #somepi submission too but I've never done any kind of video before!
I've written my own animation library over the years. If you want to program your animations, your best option is probably Manim ( www.manim.community/ ). But you can also make animations by hand, with simple tools such as powerpoint. Good luck with your video!
This is great ❤!!!
Grant cancelled #4 on October 16th, 2023. He wants to do it every other year. Too much work for him, I guess. Thanks for the video though. Expect me in your comment section asking dumb questions.
Really awesome job. Why couldn’t I watch this back when I took number theory in college? Oh yeah… RUclips hadn’t even been invented yet.
The Batman reference at 15:56 is not lost on us!
OK, wow, you really know your classics! One of my favorite quotes of all time. My hat off to you for spotting it.
Hey, numberphile discussed this once
Edit: in the form of Merten's conjecture, which is about whether or not the sum of the moebius function across the first n numbers is bounded by √n
Given a function f[x], how can you find its Dirichlet series?
Not all functions are formulated from a dirichlet series. Dirichlet series and functions are also two different things but if you regard the dirichlet series as a function, then they only classify as a small subset of all functions as they follow a specific form as seen in the video. If you are given a dirichlet series and want to see how it s composed, however, that’s a different story
May you also share the python manim code that you used for making this video. that would be really helpful
I don't use Manim, I use a custom library that isn't fit for publication.
Mobius function 😳
Wonderfully explained. Thanks.
It’s strange to me how so many maths textbooks just hand you definitions without really motivating them. It’s even stranger to me that many people don’t seem to have a problem learning that way.
It's so weird to me that the neutral element has an inverse that's not the neutral element!
It is not the inverse of the neutral element
1,0,0,0... is the neutral element
1,1,1,1... is the element taken in consideration
The function of the video is the inverse of the 1,1,1,1... sequence
Yeah! Anyway, anytime you find that a neutral element has a different inverse from itself, it signals that either it wasn’t the neutral element, or it’s not the operation under which it’s neutral (compared to the operation wrt its inverse is a different thing), or maybe even they are indeed the same modulo some natural equivalence we forgot about, or something like that.
And small errors like those do happen all the time, so it shouldn’t be any worry-despite math allows exact proofs of things by its very nature and it isn’t technically a natural science where you _absolutely need_ more varied evidence from the universe to be more certain about your theories, math still enjoys more evidence for one to become more certain in precisely the same way, so checking things from different angles (or considering particular cases and generalizations) is always a good idea even if you’re sure you have everything already proved by that point. And moreso if you’re not certain because the more insight and intuition one has, the better the road, and the more abundant are opportunities.
@@05degrees that makes sense! thank you :)
Great vid. More like Dihishlay not dirihlet
“Most integers have an even number of combinations” bro doesn’t even know squares are the same cardinality 😂
summer of math?
4th edition