Difference between revisions of "Mangoldt"

From specialfunctionswiki
Jump to: navigation, search
(Videos)
 
(5 intermediate revisions by the same user not shown)
Line 1: Line 1:
 +
__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, \\
+
\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}.
+
0, & \mathrm{otherwise}.
 
\end{array} \right.$$
 
\end{array} \right.$$
 +
 +
 +
<div align="center">
 +
<gallery>
 +
File:Mangoldtplot.png|Graph of $\Lambda$.
 +
</gallery>
 +
</div>
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
+
[[Relationship between logarithm and Mangoldt]]
<strong>Theorem:</strong> The following formula holds:
 
$$\log(n) = \displaystyle\sum_{d | n} \Lambda(d),$$
 
where $\log$ denotes the [[logarithm|natural logarithm]] and the notation $d | n$ denotes that $d$ is a divisor of $n$.
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
 
=Videos=
 
=Videos=
[https://www.youtube.com/watch?v=KTPGc4170uo Number Theory 31: Liouville and mangoldt functions]<br />
+
[https://www.youtube.com/watch?v=KTPGc4170uo Number Theory 31: Liouville and mangoldt functions] (8 January 2015)<br />
[https://www.youtube.com/watch?v=X0XJ3TuMiFc Number theory: Arithmetic functions #1]<br />
+
[https://www.youtube.com/watch?v=X0XJ3TuMiFc Number theory: Arithmetic functions #1] (27 July 2012)<br />
 +
 
 +
=References=
 +
 
 +
{{:Number theory functions footer}}
  
 
[[Category:SpecialFunction]]
 
[[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