Difference between revisions of "Mobius function"
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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>. | 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>. | ||
+ | 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) | ||
+ | |||
+ | =\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> |
Revision as of 02:46, 12 March 2012
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
\[\sum_{d|n}\mu(d)F(\frac{n}{d})=\sum_{d|n}\mu(d)\sum_{k|n/d}f(k)=\sum_{k|n}\sum_{d|n/k}\mu(d)f(k) =\sum_{k|n}f(k)\sum_{d|n/d}\mu(d)= f(n)\] (Error compiling LaTeX. Unknown error_msg)
.
The Mobius function is also closely related to the Riemann zeta function, as