Difference between revisions of "Möbius"

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=Properties=
 
=Properties=
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[[Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1]]<br />
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<strong>Theorem:</strong> If $s \in \mathbb{C}$ with $\mathrm{Re}(s) > 1$, then
 
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$$\displaystyle\sum_{n=1}^{\infty} \dfrac{\mu(n)}{n^s} = \dfrac{1}{\zeta(s)},$$
 
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where $\zeta$ is the [[Riemann zeta function]].
 
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<strong>Proof:</strong> █
 
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[[Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta]]<br />
 
[[Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta]]<br />
  

Revision as of 01:17, 22 June 2016

The Möbius function is the function $\mu$ defined by the formula $$\mu(n) = \left\{ \begin{array}{ll} 1 &; n \mathrm{\hspace{2pt}is\hspace{2pt}a\hspace{2pt}squarefree\hspace{2pt}positive\hspace{2pt}integer\hspace{2pt}with\hspace{2pt}even\hspace{2pt}number\hspace{2pt}of\hspace{2pt}prime\hspace{2pt}factors} \\ -1 &; n \mathrm{\hspace{2pt}is\hspace{2pt}a\hspace{2pt}squarefree\hspace{2pt}positive\hspace{2pt}integer\hspace{2pt}with\hspace{2pt}odd\hspace{2pt}number\hspace{2pt}of\hspace{2pt}prime\hspace{2pt}factors} \\ 0 &; n\mathrm{\hspace{2pt}has\hspace{2pt}a\hspace{2pt}square\hspace{2pt}divisor}. \end{array} \right.$$

  • Graph of $\mu$ on $[0,40]$.

  • Graph of $\mu$ on $[0,100]$.


Properties

Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1
Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta

Videos

Mobius Function Example
Möbius Function - Introduction
Möbius Function - Merten's function
Number Theory 27: Mobius function is multiplicative

References

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