  Let $\pi S(t)$ denote the argument of the Riemann zeta-function at the point
$\frac12+it$. Assuming the Riemann Hypothesis, we sharpen the constant in the
best currently known bounds for $S(t)$ and for the change of $S(t)$ in
intervals. We then deduce estimates for the largest multiplicity of a zero of
the zeta-function and for the largest gap between the zeros.
