Difference between revisions of "Mobius function"

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<cmath>\mu(n) = \begin{cases} 0 & d^2 | n, \\ (-1)^k & n = p_1p_2\cdots{p_k} .\end{cases}</cmath>
 
<cmath>\mu(n) = \begin{cases} 0 & d^2 | n, \\ (-1)^k & n = p_1p_2\cdots{p_k} .\end{cases}</cmath>
 
In addition, <math>\mu(1) = 1</math>.
 
In addition, <math>\mu(1) = 1</math>.
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The Mobius function is useful for a variety of reasons.
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First, it conveniently encodes [[Principle of Inclusion-Exclusion]].
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For example, to count the number of positive integers less than or equal to <math>n</math> and relatively prime to <math>n</math>, we have
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{| class="wikitable" style="margin: 1em auto 1em auto"
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| <math>\phi(n)</math> || <math> =n</math>
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|-
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| || <math>- \frac{n}{p_1} - \frac{n}{p_2} - \cdots - \frac{n}{p_k}</math>
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|-
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| || <math>+ \frac{n}{p_1p_2} + \frac{n}{p_1p_3} + \cdots + \frac{n}{p_{k-1}p_k}</math>
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|-
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| || <math>\vdots</math>
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|-
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| || <math>+ (-1)^k \frac{n}{p_1 p_2 \cdots p_k},</math>
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|}
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more succinctly expressed as
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<cmath>\phi(n) = \sum_{d|n} \mu(d) \frac{n}{d}.</cmath>
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One unique fact about the Mobius function, which leads to the Mobius inversion formula, is that
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<cmath>\sum_{d|n} \mu(d) = \begin{cases} 1 & n = 1, \\ 0 & \text{otherwise}. \end{cases}</cmath>
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Property 1: The function <math>\mu(n)</math> is multiplicative .
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Proof:If <math>n=1</math> or <math>p^2|n</math> for a prime <math>p</math>, we are done.Else let <math>m=\prod_{j=1}^{k} {p_j}</math> and <math>n=\prod_{i=1}^{h} {q_i}</math> where <math>gcd(m,n)=1</math>,then <math>\mu(m)\mu(n)=(-1)^{k} (-1)^{h}=(-1)^{k+h}=\mu(mn)</math>.
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Property 2:If <math>F(n)=\sum_{d|n} f(d)</math> for every positive integer <math>n</math>, then <cmath>f(n)=\sum_{d|n}\mu(d)F(\frac{n}{d})</cmath>.
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Proof:We have <cmath>\sum_{d|n}\mu(d)F(\frac{n}{d})=\sum_{d|n}\mu(d)\sum_{k|n/d}f(k)=\sum_{k|n}\sum_{d|n/k}\mu(d)f(k) =\sum_{k|n}f(k)\sum_{d|n/d}\mu(d)= f(n)</cmath> .
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The Mobius function is also closely related to the [[Riemann zeta function]], as
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<cmath>\frac{1}{\zeta(s)} = \sum \frac{\mu(k)}{n^s}.</cmath>

Revision as of 00:05, 22 September 2020

The Mobius function is a multiplicative number theoretic function defined as follows: \[\mu(n) = \begin{cases} 0 & d^2 | n, \\ (-1)^k & n = p_1p_2\cdots{p_k} .\end{cases}\] In addition, $\mu(1) = 1$.

The Mobius function is useful for a variety of reasons.

First, it conveniently encodes Principle of Inclusion-Exclusion. For example, to count the number of positive integers less than or equal to $n$ and relatively prime to $n$, we have

$\phi(n)$ $=n$
$- \frac{n}{p_1} - \frac{n}{p_2} - \cdots - \frac{n}{p_k}$
$+ \frac{n}{p_1p_2} + \frac{n}{p_1p_3} + \cdots + \frac{n}{p_{k-1}p_k}$
$\vdots$
$+ (-1)^k \frac{n}{p_1 p_2 \cdots p_k},$

more succinctly expressed as \[\phi(n) = \sum_{d|n} \mu(d) \frac{n}{d}.\]

One unique fact about the Mobius function, which leads to the Mobius inversion formula, is that \[\sum_{d|n} \mu(d) = \begin{cases} 1 & n = 1, \\ 0 & \text{otherwise}. \end{cases}\]

Property 1: The function $\mu(n)$ is multiplicative .

Proof:If $n=1$ or $p^2|n$ for a prime $p$, we are done.Else let $m=\prod_{j=1}^{k} {p_j}$ and $n=\prod_{i=1}^{h} {q_i}$ where $gcd(m,n)=1$,then $\mu(m)\mu(n)=(-1)^{k} (-1)^{h}=(-1)^{k+h}=\mu(mn)$.

Property 2:If $F(n)=\sum_{d|n} f(d)$ for every positive integer $n$, then \[f(n)=\sum_{d|n}\mu(d)F(\frac{n}{d})\]. Proof:We have \[\sum_{d|n}\mu(d)F(\frac{n}{d})=\sum_{d|n}\mu(d)\sum_{k|n/d}f(k)=\sum_{k|n}\sum_{d|n/k}\mu(d)f(k) =\sum_{k|n}f(k)\sum_{d|n/d}\mu(d)= f(n)\] .

The Mobius function is also closely related to the Riemann zeta function, as \[\frac{1}{\zeta(s)} = \sum \frac{\mu(k)}{n^s}.\]