  We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ with
upper triangular entries being independent random variables with mean zero and
unit variance. Assuming that $\max_{jk} {\mathbb E} |X_{jk}|^{4+\delta} <
\infty, \delta > 0$, it was proved in [G\"otze, Naumov and Tikhomirov,
Bernoulli, 2018] that with high probability the typical distance between the
Stieltjes transforms $m_n(z), z = u + i v$, of the empirical spectral
distribution (ESD) and the Stieltjes transforms $m_{sc}(z)$ of the semicircle
law is of order $(nv)^{-1} \log n$. The aim of this paper is to remove
$\delta>0$ and show that this result still holds if we assume that $\max_{jk}
{\mathbb E} |X_{jk}|^{4} < \infty$. We also discuss applications to the rate of
convergence of the ESD to the semicircle law in the Kolmogorov distance, rates
of localization of the eigenvalues around the classical positions and rates of
delocalization of eigenvectors.

Probability;Spectral Theory