Difference between revisions of "Mangoldt"

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(Created page with "The Mangoldt function is defined by the formula $$\Lambda(n) = \left\{ \begin{array}{ll} \log p &; n=p^k \mathrm{\hspace{2pt}for\hspace{2pt}some\hspace{2pt}prime\hspace{2pt}}p...")
 
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__NOTOC__
 
The Mangoldt function is defined by the formula
 
The Mangoldt function is defined by the formula
 
$$\Lambda(n) = \left\{ \begin{array}{ll}
 
$$\Lambda(n) = \left\{ \begin{array}{ll}
\log p &; n=p^k \mathrm{\hspace{2pt}for\hspace{2pt}some\hspace{2pt}prime\hspace{2pt}}p\mathrm{\hspace{2pt}and\hspace{2pt}integer\hspace{2pt}}k\geq 1, \\
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\log p, & n=p^k \mathrm{\hspace{2pt}for\hspace{2pt}some\hspace{2pt}prime\hspace{2pt}}p\mathrm{\hspace{2pt}and\hspace{2pt}integer\hspace{2pt}}k\geq 1, \\
0 &; \mathrm{otherwise}.
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0, & \mathrm{otherwise}.
 
\end{array} \right.$$
 
\end{array} \right.$$
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<div align="center">
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<gallery>
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File:Mangoldtplot.png|Graph of $\Lambda$.
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</gallery>
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</div>
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[[Relationship between logarithm and Mangoldt]]
<strong>Theorem:</strong> The following formula holds:
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$$\log(n) = \displaystyle\sum_{d | n} \Lambda(d),$$
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=Videos=
where $\log$ denotes the [[logarithm|natural logarithm]] and the notation $d | n$ denotes that $d$ is a divisor of $n$.
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[https://www.youtube.com/watch?v=KTPGc4170uo Number Theory 31: Liouville and mangoldt functions] (8 January 2015)<br />
<div class="mw-collapsible-content">
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[https://www.youtube.com/watch?v=X0XJ3TuMiFc Number theory: Arithmetic functions #1] (27 July 2012)<br />
<strong>Proof:</strong> █
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</div>
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=References=
</div>
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{{:Number theory functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 02:31, 28 November 2016

The Mangoldt function is defined by the formula $$\Lambda(n) = \left\{ \begin{array}{ll} \log p, & n=p^k \mathrm{\hspace{2pt}for\hspace{2pt}some\hspace{2pt}prime\hspace{2pt}}p\mathrm{\hspace{2pt}and\hspace{2pt}integer\hspace{2pt}}k\geq 1, \\ 0, & \mathrm{otherwise}. \end{array} \right.$$


Properties

Relationship between logarithm and Mangoldt

Videos

Number Theory 31: Liouville and mangoldt functions (8 January 2015)
Number theory: Arithmetic functions #1 (27 July 2012)

References

Number theory functions