Difference between revisions of "Möbius"
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[https://www.youtube.com/watch?v=yiyuu9HiXUI Möbius Function - Merten's function]<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 /> | [https://www.youtube.com/watch?v=LyyLE5ROPXA Number Theory 27: Mobius function is multiplicative]<br /> | ||
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| + | =References= | ||
| + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1}}: 24.3.1 A | ||
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Revision as of 01:16, 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
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: █
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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 24.3.1 A
