Liouville lambda

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: █