Difference between revisions of "Möbius"
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| + | =Videos= | ||
| + | [https://www.youtube.com/watch?v=9Y5xokbMBSM Mobius Function Example]<br /> | ||
| + | [https://youtu.be/zlRm1Lnz6fg?t=10 Möbius Function - Introduction]<br /> | ||
| + | [https://www.youtube.com/watch?v=yiyuu9HiXUI Möbius Function - Merten's function]<br /> | ||
| + | [https://www.youtube.com/watch?v=LyyLE5ROPXA Number Theory 27: Mobius function is multiplicative]<br /> | ||
Revision as of 00:39, 5 May 2015
The Möbius function is the function $\mu$ defined by the formula $$\mu(n) = \left\{ \begin{array}{ll} 1 &; n \mathrm{\hspace{2pt}is\hspace{2pt}a\hspace{2pt}squarefree\hspace{2pt}positive\hspace{2pt}integer\hspace{2pt}with\hspace{2pt}even\hspace{2pt}number\hspace{2pt}of\hspace{2pt}prime\hspace{2pt}factors} \\ -1 &; n \mathrm{\hspace{2pt}is\hspace{2pt}a\hspace{2pt}squarefree\hspace{2pt}positive\hspace{2pt}integer\hspace{2pt}with\hspace{2pt}odd\hspace{2pt}number\hspace{2pt}of\hspace{2pt}prime\hspace{2pt}factors} \\ 0 &; n\mathrm{\hspace{2pt}has\hspace{2pt}a\hspace{2pt}square\hspace{2pt}divisor}. \end{array} \right.$$
Properties
Theorem: If $s \in \mathbb{C}$ with $\mathrm{Re}(s) > 1$, then $$\displaystyle\sum_{n=1}^{\infty} \dfrac{\mu(n)}{n^s} = \dfrac{1}{\zeta(s)},$$ where $\zeta$ is the Riemann zeta function.
Proof: █
Videos
Mobius Function Example
Möbius Function - Introduction
Möbius Function - Merten's function
Number Theory 27: Mobius function is multiplicative
