# Tagged Questions

For questions related to Moebius inversion and its applications.

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### My attempt to follow Tatuzawa and Iseki strategy to get a bound for $\int_2^x \frac{dt}{\log t}-\pi(x)$, where $\pi(x)$ is the prime counting function

I don't know if this exercise is in the literature, where $Li(x)=\int_2^x\frac{dt}{\log t}$ is the logarithmic integral and $\pi(x)$ is the prime counting function Question. Compute a good bound ...
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### How can we have a circular sequence of $0$s and $1$s of length $d$ that are not periodic?

From: A Course in Combinatorics by van Lint / Wilson $M(d)$ is the number of circular sequences of length $d$ that are not periodic. How is this possible? If $d=1$, then the sequence $1$ is ...
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### Why does the Möbius function take its values so often in $\{0,+1,-1\}$?

The Möbius function of a locally finite poset $P$ is defined on its intervals $[x,y] \subseteq P$ recursively by $$\mu([x,x])=1$$ $$\forall x < y : \mu([x,y])=-\sum_{x \leq z < y} \mu([x,z])$$ ...
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### Can you analyze this identity involving the sum of divisors function and $rad(n)=\prod_{p\mid n}p$?

Let $\sigma(n)$ the sum of divisors function, then by Mobius inversion formula $$\sigma(n)=n-\sum_{\substack{d\mid n,d<n}}\sigma(d)\mu\left(\frac{n}{d}\right),$$ and since this function is ...
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### Why does the inversion of the unit circle centered at $-i$ give the horizontal line $y = \frac{1}{2}$?

Inverting the circle centered at $-i$ with radius $1$, gives the horizontal line $y = \frac{1}{2}$, but why does the line have to be horizontal? Why not another straight line passing through the ...
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### Calculating Moebius inversion of a poset with a minimum element.

If P is a poset that has a minimum element. We let x be an element of P that covers 1 single element y.Assume that y is not the minimum element, how do I prove that μ(minimum element,x) = 0? So ...
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### How to compute the Mobius function

I have no clue how to begin this problem. It involves computing the Mobius inversion function $\mu$. This problem comes from Stanley's $\textit{Enumerative Combinatorics}$, vol 1, problem 70, Chapter ...
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### A question about step in the proof of Selberg's formula

Recently I've found the following paper, discussing and proving Selberg's symmetry formula: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/Balady.pdf My question concerns proofs of ...
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### Question - Möbius inversion formula

I need your help in the next question: Prove directly from the definition the Möbius inversion formula. (Möbius function defined as follows: μ(n) = 1 if n is a square-free positive integer with ...
for $x \ge 2863:$ $$\ln\left(\left\lfloor\frac{x}{6}\right\rfloor!\right) < \sum_{k=5}^{\infty}-\mu(k)\ln\left(\left\lfloor\frac{x}{k}\right\rfloor!\right)$$ I've written a java application which ...