Difference between revisions of "Möbius"

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The Möbius function is the function $\mu$ defined by the formula
 
The Möbius function is the function $\mu$ defined by the formula
 
$$\mu(n) = \left\{ \begin{array}{ll}
 
$$\mu(n) = \left\{ \begin{array}{ll}
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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 />
<strong>Theorem:</strong> 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]].
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
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</div>
 
 
 
 
[[Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta]]<br />
 
[[Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta]]<br />
  
 
=Videos=
 
=Videos=
[https://www.youtube.com/watch?v=9Y5xokbMBSM Mobius Function Example]<br />
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[https://youtu.be/zlRm1Lnz6fg?t=10 Möbius Function - Introduction (4 September 2007)]<br />
[https://youtu.be/zlRm1Lnz6fg?t=10 Möbius Function - Introduction]<br />
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[https://www.youtube.com/watch?v=yiyuu9HiXUI Möbius Function - Merten's function (4 September 2007)]<br />
[https://www.youtube.com/watch?v=yiyuu9HiXUI Möbius Function - Merten's function]<br />
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[https://www.youtube.com/watch?v=9Y5xokbMBSM Mobius Function Example (17 November 2012)]<br />
[https://www.youtube.com/watch?v=LyyLE5ROPXA Number Theory 27: Mobius function is multiplicative]<br />
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[https://www.youtube.com/watch?v=LyyLE5ROPXA Number Theory 27: Mobius function is multiplicative (8 January 2015)]<br />
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[https://www.youtube.com/watch?v=Vsib1v5vfkc Möbius Inversion of $\zeta(s)$ (3 July 2016)]<br />
  
 
=References=
 
=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
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* {{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}}: $24.3.1 \mathrm{I}.A.$
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{{:Number theory functions footer}}
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 23:55, 8 December 2016

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


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

Möbius Function - Introduction (4 September 2007)
Möbius Function - Merten's function (4 September 2007)
Mobius Function Example (17 November 2012)
Number Theory 27: Mobius function is multiplicative (8 January 2015)
Möbius Inversion of $\zeta(s)$ (3 July 2016)

References

Number theory functions