# Questions tagged [mobius-inversion]

For questions related to Möbius inversion and its applications.

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### Circle Inversion on the Line at Infinity [closed]

We know that a circle inversion of point ($x_1,x_2$) about a circle of radius r is given by: Wiki Blurb ($x_1,x_2$) $\implies$ ($\frac{rx_1}{\lambda^2 }$,$\frac{rx_2}{\lambda^2 }$) Where $\lambda^2$ = ...
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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: $$z = T^{-1}=\frac{w-1}{w+1}$$ ...
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The expanded partial sums of the Möbius inverse of the Harmonic numbers have two out of three properties in common with this set of linear programming problems: $$\begin{array}{ll} \text{minimize} &... 2answers 85 views ### How to write \zeta(s)^3 as a function of the generalized Mobius function? Popovici (1963) (see link), created a way to extend the Mobius function, \mu(n), to the complex plane. The Mobius \mu(n) function is such that: \frac{1}{\zeta{(s)}}=\sum_{n=1}^{\infty}\frac{\mu(... 1answer 62 views ### Simplifying expression Mobius Function Can anybody help me simplify this expression using Mobius Inversion Formula or any other result in order to calculate F(3500) in a simple way??$$F(n)=\sum_{d\mid n} \mu(d)d$$1answer 44 views ### Two sequence relations are equivalent. For two sequences of complex numbers \{a_0, a_1, ..., a_n, ...\} and \{b_0, b_1, ..., b_n, ...\}. Show that the following relations are equivalent: a_n = \sum_{k=0}^n b_k \iff b_n ... 1answer 108 views ### Number of words with length n and symbols \{1,2,3\} which are not equal to any of their cyclic shifts, with Möbius Inversion. Let t_{n} be the number of words with length n and symbols \{1,2,3\} which are not equal to any of their cyclic shifts. For example, the word 1232 has cyclic shifts 2123, 3212, 2321, none of ... 1answer 80 views ### Easy way to prove property of prime indicator Let \mu be the Möbius function, and let \nu(n) be the number of distinct prime factors of n. Then we can define p = \mu * \nu, i.e.$$ p(n) = \sum_{d \mid m} \mu(d) \nu(n/d). $$An exercise in ... 1answer 487 views ### Proof of Mobius inversion formula - the other direction The Mobius inversion formula states that if we define f as$$f(m)=\sum_{d|m}g(d)$$then$$g(m)=\sum_{d|m}f\left(\frac md\right)\mu(d)$$We know that the other direction is also true: if we define ... 1answer 103 views ### How to show that \sum_{d\mid n} \gcd(d,k) \mu(n/d)=0? Fix k\in \Bbb{N} such that k > 1. Define f_k(n)=\gcd(n,k) for \forall\ n \geq 1. Let \mu denote the Möbius function. Notice that$$\sum_{d\mid n}f_k(d)\mu\left(\frac{n}{d}\right)=0 for ...
Let $ABC$ be a curved triangle on a plane, whose side $AB$, $BC$ and $CD$ are arcs of circles $S_1$, $S_2$ and $S_3$ passing though a point $D$ (i.e. $S_1∩S_2∩S_3 = D$, $D≠A$, $D≠B$, $D≠C$). Assume ...
### $f(n) = \Sigma_{d|n} \mu(n/d)F(d)$
The question says: If $F(n) = \Sigma_{d|n} f(d)$ for every positive integer $n$, prove that $f(n) = \Sigma_{d|n} \mu(n/d)F(d)$. What I know so far is that divisors of $n$ can be paired together. ...