# Difference between revisions of "Mobius function"

Line 30: | Line 30: | ||

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>. | 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>. | ||

− | 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) | + | 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> . |

− | |||

− | =\sum_{k|n}f(k)\sum_{d|n/d}\mu(d)= f(n)</cmath> . | ||

The Mobius function is also closely related to the [[Riemann zeta function]], as | The Mobius function is also closely related to the [[Riemann zeta function]], as | ||

<cmath>\frac{1}{\zeta(s)} = \sum \frac{\mu(k)}{n^s}.</cmath> | <cmath>\frac{1}{\zeta(s)} = \sum \frac{\mu(k)}{n^s}.</cmath> |

## Latest revision as of 00:05, 22 September 2020

The Mobius function is a multiplicative number theoretic function defined as follows: In addition, .

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 and relatively prime to , we have

more succinctly expressed as

One unique fact about the Mobius function, which leads to the Mobius inversion formula, is that

Property 1: The function is multiplicative .

Proof:If or for a prime , we are done.Else let and where ,then .

Property 2:If for every positive integer , then . Proof:We have .

The Mobius function is also closely related to the Riemann zeta function, as