# Questions tagged [mobius-inversion]

For questions related to Möbius inversion and its applications.

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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 ...
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### Möbius inversion for categories instead of directed graphs

In Tom Leinster, The Euler Characteristic Of A Category, the author generalizes the notion of Möbius Inversion for posets to finite categories. This violates the principle of equivalence. A possible ...
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### Mobius inversion on the partition lattice

For some $n \in \mathbb N$, let $(\Pi_n, \le)$ be the poset of partitions of the set $\{1, 2, \dots, n\}$, where two partitions $\pi, \rho \in \Pi_n$ have the relation $\pi \le \rho$ if $\pi$ is a ...
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### Correspondence between $k$-ary Lyndon words and $(k-1)$-ary Lyndon words without repetitions

At Counting Lyndon words with no adjacent character repeats, it turned out that for $n\ge3$ the number of $k$-ary Lyndon words of length $n$ without adjacent identical letters is the number of $(k-1)$-...
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### Proof of an identity concerning the prime $\zeta$ function

I have to prove the following identity: let $P(s)=\sum_p\frac{1}{p^s}$, for $Re(s)>1$, then $$P(s)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log(\zeta(ns)).$$ I proved that \...
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### Question on proof of relationships between $f(s)=\frac{s}{s+1}$ and the analytic Harmonic number function $H(s)$

This question assumes the following definitions where $s\in\mathbb{C}$. (1) $\quad f(s)=\frac{s}{s+1}$ (2) $\quad H(s)=\psi(s+1)+\gamma\qquad\text{(analytic harmonic number function)}$ Question: ...
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### Inversive distance between concentric circles

I found an explication about Inversive distance in Coxeter book. But, i don't really understand some things in its explication. I can't imagine the meaning of: ''This inquiry almost forces us to ...
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### Construction of inverse point in sphere

i would like to know if my construction of inverse point of A in sphere, A' is right. First of all i constructed a line through pole $S$ and $A$, then i drew a line through $N$ perpendicular to $AS$. ...
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Prove by Mobius inversion formula if $\frac{n}{\phi(n)}=\sum_{d\mid n} f(d)$ then $f(d)=\frac{\mu^2(d)}{\phi(d)}.$