  We establish quantitative bounds on the $U^k[N]$ Gowers norms of the M\"obius
function $\mu$ and the von Mangoldt function $\Lambda$ for all $k$, with error
terms of shape $O((\log\log N)^{-c})$. As a consequence, we obtain quantitative
bounds for the number of solutions to any linear system of equations of finite
complexity in the primes, with the same shape of error terms. We also obtain
the first quantitative bounds on the size of sets containing no $k$-term
arithmetic progressions with shifted prime difference.
