# Questions tagged [mobius-inversion]

For questions related to Möbius inversion and its applications.

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### Relation between $f$ and $g$ satisfying $\sum_{r}^N f(x^r) = g(x)$ without mobius inversion?

Let's say for a particular function I have $f$: $$f(x) + f(x^2)+ f(x^3)+ \dots +f(x^N) = g(x)$$ We start by considering the integral: $$I_f = \int_0^{b} f (e^{-\frac{1}{x}}) dx$$ Using asymptotics ...
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### Prove inversion formula involving binomial coefficients

Let's say that we have such an equation: $$f_k = \sum_{i=0}^{k}{k \choose i}g_i$$ Prove that in that case we can express $g_k$ like this: $$g_k = \sum_{i=0}^{k}(-1)^{k-i}{k \choose i}f_i$$ In order to ...
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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 $AS$. ...
Prove by Mobius inversion formula if $\frac{n}{\phi(n)}=\sum_{d\mid n} f(d)$ then $f(d)=\frac{\mu^2(d)}{\phi(d)}.$