Difference between revisions of "Liouville lambda"
From specialfunctionswiki
| Line 24: | Line 24: | ||
</div> | </div> | ||
</div> | </div> | ||
| + | |||
| + | {{:Number theory functions footer}} | ||
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Latest revision as of 06:35, 22 June 2016
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: █
