The Paillier CryptoSystem has a public key that $(g,n)$ and the private key which can be exclusive to $\lambda$, where the decryption scheme is:
$m = L(c^\lambda \bmod n^2)/L(g^\lambda \bmod n^2) \bmod n$
Since $1/L(g^\lambda \bmod n^2)$ is fixed and always needed for decryption, it is usually computed once and denoted as $\mu$.
What information does $\mu$ leaks about $\lambda$? Because at the end of the day, even if I have $\mu$, I cannot decrypt. i.e. Can I get $\lambda$ from $\mu$?
A Side Note on the way $\mu$ is constructed, that I think proves the correctness of the assumption:
\begin{align} g &= (1+n)^\alpha \cdot \mathcal{B}^n \pmod{n^2} & & \text{$g$ in the $n^{\text{th}}$ root form} \\ g^\lambda &= (1+n)^{\alpha\lambda} \cdot \mathcal{B}^{n\lambda} \pmod{n^2} & &\text{so base on carmichael's theorem} \\ g^\lambda &= (1+n)^{\alpha\lambda} \pmod{n^2} & & \text{again, based on $n^{\text{th}}$ root rule}\\ g^\lambda &= 1+n\alpha\lambda \pmod{n^2}& & \\ L(g^\lambda) &= \alpha\lambda \pmod{n^2}& &\\ \mu &= 1/\alpha\lambda \pmod{n^2} & \end{align} So, since it is impossible to get $\alpha$ given $g$, the main complexity of the encryption scheme itself, and the last equation is a function of two variable, and there is no way to find either variable.