The Liouville function is defined by $\lambda (1) = 1$ and $\lambda (n) = (-1)^{k_1 + k_2 + \cdots + k_r}$ , if the prime factorization of $n > 1$ is $n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$ (where each $p_i$ is positive). This function is completely multiplicative and satisfies the identity