# Questions tagged [mobius-inversion]

For questions related to Möbius inversion and its applications.

117 questions
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### How to show that $\sum_{d\mid n} \gcd(d,k) \mu(n/d)=0$?

Let $k\in \Bbb{N}$ and $k>2$. Fix $k$, define $f_k(n)=\gcd(n,k)$ for $\forall\ n>1$. Let $\mu$ denote the Möbius function. Notice that $$\sum_{d\mid n}f_k(d)\mu\left(\frac{n}{d}\right)=0$$ for ...
1answer
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### Mobius transformation producing a curved triangle with 3 intersecting circles

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 ...
2answers
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### $f(n) = \Sigma_{d|n} \mu(n/d)F(d)$

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. ...
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### Möbius inversion across all natural numbers (no divisors)

EDITED TO ACCOMMODATE COMMENTS: I'm trying (self-taught) to understand more about Möbius inversion. Take two arithmetic functions $f$ and $g$ defined by $$g(n)=\sum_{d|n}f(d)$$ (Presumably $d$ ...
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1answer
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### Convergence of the limit of a series involving Möbius and Floor functions

Studying and working with some problems involving the Möbius function, I (erroneously and somewhat randomly) found the following series/limit. I am curious about it, since I don't know wether it ...
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### Möbius inversion exercise

I have been given the following exercise: Get a function $G(x)$ so that, being $x$ a positive integer, $$x+\sqrt{x}= \sum^x_{n=1} \mu(n) G(\frac{x}{n})$$ The result does not have to be exact, but ...
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### Rate of convergence of a series

What would be the rate of convergence of: $$\sum^N_n \frac{\mu(n) \log(n)}{n}$$ ? I know that as $N \to \infty$ the series approaches to $-1$, but I am not able to get how fast it does converge ...
1answer
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### How to show this equality using Mobius function?

Let $n$ be a natural number greater than $1$. Prove the following: $$\sum_{d|n} \mu(d) \sigma(d)=(-1)^rp_1...p_r$$where $p_1,...,p_r$ are the distict prime factors of $n$ and $\mu$ is the Mobius ...
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### Prove $\sum_{d|n} \frac{\Phi(d)}{d} = \prod_{i=1}^r (1 + a_i - \frac{a_i}{p_i})$

I want to prove $\sum_{d|n} \frac{\Phi(d)}{d} = \prod_{i=1}^r (1 + a_i(1 - \frac{1}{p_i}))$, where $\Phi(n)$ is the Euler phi function and given the prime factorisation $n = \prod_{i=1}^r p_i^{a_i}$. ...
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### Fractals and Kleinian groups - Rendering the limit set

I am talking about this: I recently read the book "The fractal geometry of nature" by Benoit B. Mandelbrot. There was one particular fractal I found very beautiful: A limit set of some group of ... ...
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### Generalization of Möbius Inversion

I understand the proof of the Möbius inversion formula by moving the inner summation of $$\sum_{d|n}\mu(n/d)\sum_{k|d}g(k) \\$$ outside, giving: $$\sum_{k|n}g(k)\sum_{q|n/k}\mu(n/kq) = g(n)$$ ...
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