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#numbertheory #moebius #dirichlet #somepi
In number theory, the Moebius function allows us to decompose complicated functions into simpler parts. The definition of this function can be difficult to understand, so we flesh it out one step at a time. We start with the Dirichlet convolution, we look at its properties, and finally we look for inverses of number sequences. One of those inverses is the Moebius function. We understand how it works by looking at a Hasse diagram.
To learn more about the Moebius function and other topics in number theory, here are some very good links to get you started:
[[3B1B 1]] https://www.youtube.com/watch?v=KuXjwB4LzSA
A beautiful explanation of 2-dimensional convolutions of images. This is an extension of the 1-dimensional convolution of sequences of numbers that we talk about in our video.
[[WIKI 1]] https://en.wikipedia.org/wiki/M%C3%B6bius_function
Many more details about the Moebius function.
[[WIKI 2]] https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula
This wikipedia page talks about the Moebius transform, and how you can invert it using the Moebius function.
[[WIKI 3]] https://en.wikipedia.org/wiki/Dirichlet_convolution
More examples of Dirichlet convolutions.
[[WIKI 4]] https://en.wikipedia.org/wiki/Modular_form
Modular forms are complex functions that play a central role in number theory. We may make a few videos about this in the future.
[[VER 1]] https://www.youtube.com/watch?v=Zrv1EDIqHkY
More information about perfect numbers and the sigma function.
0:00 Introduction
1:14 Polynomial multiplication
3:29 Dirichlet convolution
7:14 Examples of Dirichlet convolution
11:00 Neutral element and inverses
16:01 Hasse diagram and definition of mu
This video is published under a CC Attribution license
( https://creativecommons.org/licenses/by/4.0/ )