# Liouville function

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 the

 $\sum_{d|n}\lambda(d)=\begin{cases}1&\text{if n=m^{2} for some integer m}\\ 0&\text{otherwise,}\end{cases}$

where the sum runs over all positive divisors of $n$.

Title Liouville function LiouvilleFunction 2013-03-22 11:47:09 2013-03-22 11:47:09 KimJ (5) KimJ (5) 12 KimJ (5) Definition msc 20G10 msc 11A25 msc 81-00