I can show that $\lambda (n)=i^{\tau(n^{2})-1}$, where $\lambda (n)$ is the Liouville function, $\tau(n)$ is the divisor function, and $i$ is the imaginary unit.
My question is as stated, and what is the best it could be currently? Does the divisor function make it any easier?
I have little to no experience with computation complexity calculation.
Usually, the definition is $\lambda (n)=(-1)^{\Omega(n)} $ where $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity.
If $n =\prod p_n^{k_i} $ then $\Omega(n) =\sum k_i $.
Therefore the time to compute $\Omega(n)$ is at most the time needed to factor $n$, so you can look at articles like this to get an idea of what is currently known:
https://en.wikipedia.org/wiki/Integer_factorization