# Questions tagged [mobius-inversion]

For questions related to Möbius inversion and its applications.

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### Problems about inversion map

Let $\tau_{\omega}$ be the inversion with respect to circle $\omega$ and $\omega$ is the unit circle in $\mathbb{E}^2$ with centre $O=(0,0)$. There are three cases: a. Find $\tau_{\omega}(\ell)$ where ...
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### Prime numbers as zeros of polynomials from the determinant of the form $P(x,N)=|\log(x) \gcd (n,k)-\phi (n)\log(\gcd (n,k))|$

Let $f(n)$ be some arbitrary sequence $\log(n),n,n^2,\dots$ $$f(n)=\log(n),n,n^2,\dots$$ and $$a(n)=\frac{\sum\limits_{d \mid n} f(d) \mu \left(\frac{n}{d}\right)}{\phi (n)}$$ and construct the ...
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### Proof of the dual Möbius Inversion Formula

The Dual Möbius Inversion Formula: Let $\mathcal D$ be a divisor closed set of natural numbers (i.e., if $d\in \mathcal D$ and $c \mid d$, then $c\in \mathcal D$.) Let $f$ and $g$ be two complex-...
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### Applications of the mobius inversion in algebraic or arithmetic geometry

I was wondering if there were any generalisations or applications of the ideas of the Möbius inversion formula to more modern areas of mathematics such as algebraic or arithmetic geometry. I know they ...
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### Characterization of Möbius-monotonicity

We say that an arithmetic function $f:\mathbb{Z}^+\to\mathbb{C}$ is Möbius-monotone if $\forall n\geq1:(\mu*f)(n)\geq0$ (where $*$ denotes de Dirichlet convolution), i.e. if there exists a non-...
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### Applying the Möbius inversion formula to $f(n) = \sum_{p\mid n}g(p)$?

Is there a specific technique that exists to reducing summation over divisors to prime divisors? Specifically I am interested in applying the Möbius inversion formula to an arithemetic function of the ...
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### Product of all reduced residues in relation with the function $\mu$: $\prod_{1\leq a\leq n,(a,n)=1}a=n^{\varphi(n)}\prod_{d|n}(d!/d^d)^{\mu(n/d)}$

We know that for any arithmetic function $f$ the Mobius inversion formula gives its inversion. Hence $$F(n)=\prod_{d|n}f(d)\implies f(n)=\prod_{d|n} F(n/d)^{\mu(d)}.$$ The above statement can be ...
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### Task , given matrices, it is necessary to make the matrices "F" and "G" according to the scheme

enter image description here ( 2 ); B = (-2); C = (-1); D = = G₁ Task 3. Given matrices A = C= 1): D = (-2 1 2). 1) according to the schemes, make a mat matrices F and G. Find the products of matrices ...
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### Summatory function of Euler-phi

Let $F(n) = \sum_{d^2|n} \phi(d)$. We must show that if $F(1) = 1$, and if $n>1$ factors as $n=p^{a_1}_1p^{a_2}_2...p^{a_m}_m$, then $$F(n)=\prod_{i=1}^{m} p^{[a_i/2]}_i.$$ If I understood ...