  We consider ensembles of Wigner matrices, whose entries are (up to the
symmetry constraints) independent and identically distributed random variables.
We show the convergence of the Stieltjes transform towards the Stieltjes
transform of the semicircle law on optimal scales and with the optimal rate.
Our bounds improve previous results, in particular from [22,10], by removing
the logarithmic corrections. As applications, we establish the convergence of
the eigenvalue counting functions with the rate $(\log N)/N$ and the rigidity
of the eigenvalues of Wigner matrices on the same scale. These bounds improve
the results of [22,10,23].
