# Questions tagged [mobius-inversion]

For questions related to Möbius inversion and its applications.

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### Can lines intersect twice?

In inversion,we extend the euclidean plane by adding a single point at infinity which lies on all the lines.But doesn't that mean lines can intersect twice?I mean non parallel lines already intersect ...
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### Understanding a detail in the inclusion-exclusion Möbius Inversion proof in Introductory Combinatorics

In Introductory Combinatorics, by Brualdi, we have an explication on the Möbius Inversion. There's a line that I'm having a hard time believing, and would appreciate some explanation: Let $n$ be a ...
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### Dirichlet Self-Convolution Inversion

I am interested in finding out a method to invert Dirichlet selfconvolution. In math expressions it means: Find out $a$ once $b=a*a$ is known So a kind of squareroot of the Dirichlet product. I ...
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### Möbius function and Posets

As usual, $\mathbb{C}$ denote the field of complex numbers. Let $\mu \in I_{\mathbb{C}}(P)$ (the $\mathbb{C}$ incidence-algebra of $P(X, \leq)$ a poset). I am asked to show the following are ...
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### $\Phi_{2^n}(x) = x^{2^{n-1}}+1$ by Mobius Inversion

I want to prove that $\Phi_{2^n}(x) = x^{2^{n-1}}+1$. This is not hard to do by using recursion. However, I want to know if there's a simpler way to do this using the Mobius Inversion formula, which ...
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### Squarefree integers and floor function, Möbius function

On page 40, exercise 44 of Introduction to Analytic and Probabilistic Number Theory by Tenenbaum: Show that any integer $n\ge1$ can be uniquely decomposed as $n = qm^2$ , where $q$ is squarefree. ...
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### Number of subsets of size $k$ with a given GCD

Given a set $S = \left\{1, 2, \ldots,N \right\}$ of positive integers, we want to count the number of subsets of size $k$ with the GCD of all elements (henceforth referred to as the GCD of the set) ...
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### Need help with Möbius function

Suppose I need to find total subset of numbers of length $K$ in range $1$ to $N$ such that their $\gcd$ is $g$. How can I utilize Möbius function for that. So approach is we can choose $K$ numbers ...
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### Arbelos - Pappus chain radius of tangent circles

I would like to determine radius' of pappus chain circles which are tangent to Arbelos. Below I will put some photos from one book that showed how to determine it. The only thing that i don't ...
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### which domain does this Möbius map to $\Re(w) > 0$?

Given the following Möbius: $$w = T(z) = \frac{1+z}{1-z}$$ How could I find the domain of $Z$ which $T$ maps to $\{\Re(w)>0\}$? I tried to inverse $T$ and got: $$z = T^{-1}=\frac{w-1}{w+1}$$ ...