This question assumes the following:
- $\nu(n)$ is the number of distinct primes in the factorization of $n$,
- $\omega(n)$ is the number of prime factors counting multiplicities in the factorization of $n$,
- $\lambda(n)$ is defined to be $(-1)^{\omega(n)}$, and
- $\delta_{i,j}=\cases{0,& $i\ne j$ \\ 1,& $i=j$}\quad$ is the Kronecker delta function.
Question: Has the following formula been proven (or disproven) and if not, can it be?
(1) $\quad\sum\limits_{d|n} 2^{\nu(d)+\nu\left(\frac{n}{d}\right)}\lambda(d)=\delta_{n,1}=\cases{0,& $n\ne 1$ \\ 1,& $n=1$}$
The case $n=1$ is trivial, so my interest is a proof for the case $n>1$.
The Liouville function $\lambda(n)$ is closely related to the Möbius function $\mu(n)$, and formula (1) above seems somewhat analogous to formula (2) below.
(2) $\quad\sum_\limits{d|n}\mu(d)=\delta_{n,1}$
$2^{\nu(n)} \lambda(n),2^{\nu(n)}$ are multiplicative functions so $$ f(n)=\sum_{d | n}\lambda(d) 2^{\nu(d)}2^{\nu(n/d)} $$ is multiplicative ie. $$ \begin{split} f(n) &= \prod_{p^k \| n} f(p^k)=\prod_{p^k \| n} \left(\sum_{d | p^k}\lambda(d) 2^{\nu(d)}2^{\nu(p^k/d)}\right)\\ &=\prod_{p^k \| n} \left(\sum_{l=0}^k \lambda\left(p^l\right) 2^{\nu(p^l)}2^{\nu(p^k/p^l)}\right)\\ &= \prod_{p^k \| n} \left[2+\left(\sum_{l=1}^{k-1} (-1)^l 4\right) + 2(-1)^k \right] \end{split} $$