  We investigate one dimensional SU$(N)$ Hubbard chains at zero temperature,
which can be emulated with ultracold alkaline earth atoms, by using the density
matrix renormalization group (DMRG), Bethe ansatz (BA), and bosonization. We
compute experimental observables and use the DMRG to benchmark the accuracy of
the Bethe ansatz for $N>2$ where the BA is only approximate. In the worst case,
we find a relative error $\epsilon \lesssim 4%$ in the BA ground state energy
for $N \leq 4$ at filling 1/N, which is due to the fact that BA improperly
treats the triply and higher occupied states. Using the DMRG for $N \leq 4$ and
the BA for large $N$, we determine the regimes of validity of strong- and
weak-coupling perturbation theory for all values of $N$ and in particular, the
parameter range in which the system is well described by a SU$(N)$ Heisenberg
model at filling 1/N. We find this depends only weakly on $N$. We investigate
the Berezinskii-Kosterlitz-Thouless phase transition from a Luttinger liquid to
a Mott-insulator by computing the fidelity susceptibility and the Luttinger
parameter $K_\rho$ at 1/N filling. The numerical findings give strong evidence
that the fidelity susceptibility develops a minimum at a critical interaction
strength which is found to occur at a finite positive value for $N>2$.
