What is the Moebius function?  

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

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!

I loved the generatingfunctionology chapter on this. I actually love that book

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)

Brilliant, one of the best entries I've seen after all these years.

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).

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

Great video.

Just wow. Amazing. Thanks again

💛 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!

Mobius function 😳

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.

Given a function f[x], how can you find its Dirichlet series?

It's so weird to me that the neutral element has an inverse that's not the neutral element!

Great vid. More like Dihishlay not dirihlet

summer of math?