Difference between revisions of "Liouville lambda"
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=Properties= | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> For every $n \geq 1$ | ||
| + | $$\displaystyle\sum_{d | n} \lambda(d) = \left\{ \begin{array}{ll} | ||
| + | 1 &; n \mathrm{\hspace{2pt}is \hspace{2pt} a \hspace{2pt} square} \\ | ||
| + | 0 &; \mathrm{otherwise}, | ||
| + | \end{array} \right.$$ | ||
| + | where $d | n$ denotes that the sum is over all [[divisor|divisors]] $d$ of $n$. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
<strong>Theorem:</strong> The following formula holds: | <strong>Theorem:</strong> The following formula holds: | ||
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where $\zeta$ denotes the [[Riemann zeta function]]. | where $\zeta$ denotes the [[Riemann zeta function]]. | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
| − | <strong>Proof:</strong> | + | <strong>Proof:</strong> █ |
</div> | </div> | ||
</div> | </div> | ||
Revision as of 04:10, 30 April 2015
The Liouville function is defined by the formula $$\lambda(n) = (-1)^{\Omega(n)},$$ where $\Omega(n)$ indicates the number of prime factors of $n$, counted with multiplicity.
Properties
Theorem: For every $n \geq 1$ $$\displaystyle\sum_{d | n} \lambda(d) = \left\{ \begin{array}{ll} 1 &; n \mathrm{\hspace{2pt}is \hspace{2pt} a \hspace{2pt} square} \\ 0 &; \mathrm{otherwise}, \end{array} \right.$$ where $d | n$ denotes that the sum is over all divisors $d$ of $n$.
Proof: █
Theorem: The following formula holds: $$\dfrac{\zeta(2s)}{\zeta(s)}=\displaystyle\sum_{n=1}^{\infty} \dfrac{\lambda(n)}{n^s},$$ where $\zeta$ denotes the Riemann zeta function.
Proof: █
