Questions tagged [mobius-function]

Questions on the Möbius function μ(n), an arithmetic function used in number theory.

269 questions
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Prove the sum of the Mobius function over monic polynomials of degree $n$ is $0$ if $n > 1$

Let $\mu(m)$ be the Möbius function on monic polynomials in $\mathbb{F}_q[x]$ ($q$ is power of prime) where $\mu(m) = 0$ if $m$ is not square-free and $\mu(m) = (-1)^k$ if $m$ is square-free and can ...
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The number of ways, $sf_2(n)$, to express an integer as the sum of two square-free integers.

It is well known that $$\sum_{n\leq x}\mid \mu(n) \mid \sim \frac{6}{\pi^2}x\left(1+o(1)\right) \text{, } x \to \infty$$ From here it follows that every sufficient large integer may be expressed as ...
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Name of poset operation involving Möbius function

If we have a finitary poset $P$ with a $\hat{0}$ (least element), and we want to compute the Möbius function for all elements $x$, as in $\mu(\hat{0},x)$, it wouldn't affect any of our computations to ...
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Upper bound for the reciprocal of a sum involving Möbius function

For $n$ an integer greater than $6$ , let $Q(n)=\prod_{p\leq\sqrt{2n-3}}p$. Which upper bound in terms of $n$ can we get for $(\sum_{d\mid Q(n)}\frac{\mu(d)}{d})^{-1}$?
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How can one calculate the Möbius function $\mu(a_1,a_i)$ for all $i \in \{1, …, 10\}$ of this poset?

I've seen this partially ordered set in our combinatorics script and it says that it is obvious how to calculate the möbius function $\mu(a_1,a_i)$ for all $i \in \{1, ..., 10\}$. Here's the Hasse ...
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Degree of a formal power series involving Mobius function

I am reading Enumerative Combinatorics by Richard Stanley, and I came across the following expression: $(1-x^n)^{\frac{-\mu(n)}{n}}$, where $\mu(n)$ is the usual Mobius function from number theory. ...
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Upper bound of $\sum_{n\leq x}f(n)$ where $f(n)=\sum_{r=1}^{n-1}\mu(r)\mu(n-r)$
$Cx^2$ is a trivial bound by just counting the total number of terms in these sums. From here I have attempted to use  \sum_{n\leq x} \mid \mu(n) \mid =\frac{6}{\pi^2}x(1+o(1)) \text{, } x \to \...