what is number of invertible matrix m*m on$ z_n$? ‎


  ‎‎please help me what is number of invertible matrix $‎m*m$‎‎‎ on Group $\mathbb{z}_n$ ?‎‎‎, assuming we know‎ this number in $\mathbb{Z}_p \quad$ (‎p is prime‎‎) is   $‎(p^{n}-1)(p^{n}-p) \cdots (p^n-p^{n-1})‎‎‎‎$
Assume you know the number of invertible $m \times m$ matrices over $\mathbb{Z}_{p^k}$ for $p$ prime. Call this number $N(p^k)$.

By the Chinese Remainder Theorem, $N(\prod_{i=1}^l p_i^{k_i}) = \prod_{i=1}^l N(p_i^{k_i})$ because a matrix is invertible iff every component in its CRT decomposition is invertible. For $m = 1$, $N$ is just the $\phi$ function.

Finding $N(p^k)$ is another matter.