Difference between revisions of "Möbius"
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0 &; n\mathrm{\hspace{2pt}has\hspace{2pt}a\hspace{2pt}square\hspace{2pt}divisor}. | 0 &; n\mathrm{\hspace{2pt}has\hspace{2pt}a\hspace{2pt}square\hspace{2pt}divisor}. | ||
\end{array} \right.$$ | \end{array} \right.$$ | ||
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| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Mobiusmuplot,n=0..100.png|Plot of $\mu$ for $n=0,\ldots,100$. | ||
| + | File:Mobiusmuplot,n=0..500.png|Plot of $\mu$ for $n=0,\lodts,500$. | ||
| + | </gallery> | ||
| + | </div> | ||
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=Properties= | =Properties= | ||
Revision as of 21:44, 16 August 2015
The Möbius function is the function $\mu$ defined by the formula $$\mu(n) = \left\{ \begin{array}{ll} 1 &; n \mathrm{\hspace{2pt}is\hspace{2pt}a\hspace{2pt}squarefree\hspace{2pt}positive\hspace{2pt}integer\hspace{2pt}with\hspace{2pt}even\hspace{2pt}number\hspace{2pt}of\hspace{2pt}prime\hspace{2pt}factors} \\ -1 &; n \mathrm{\hspace{2pt}is\hspace{2pt}a\hspace{2pt}squarefree\hspace{2pt}positive\hspace{2pt}integer\hspace{2pt}with\hspace{2pt}odd\hspace{2pt}number\hspace{2pt}of\hspace{2pt}prime\hspace{2pt}factors} \\ 0 &; n\mathrm{\hspace{2pt}has\hspace{2pt}a\hspace{2pt}square\hspace{2pt}divisor}. \end{array} \right.$$
Plot of $\mu$ for $n=0,\ldots,100$.
Plot of $\mu$ for $n=0,\lodts,500$.
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: █
Theorem
The following formula holds: $$P(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)}{k} \log \zeta(kz),$$ where $P$ denotes the Prime zeta function, $\mu$ denotes the Möbius function, $\log$ denotes the logarithm, and $\zeta$ denotes the Riemann zeta function.
Proof
References
Videos
Mobius Function Example
Möbius Function - Introduction
Möbius Function - Merten's function
Number Theory 27: Mobius function is multiplicative
