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 /> | ||
| + | [https://www.youtube.com/watch?v=Vsib1v5vfkc Möbius Inversion of $\zeta(s)$]<br /> | ||
=References= | =References= | ||
Revision as of 18:07, 5 July 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
Möbius Inversion of $\zeta(s)$
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $24.3.1 \mathrm{I}.A.$
