A note on S(T) and the zeros of the Riemann zeta-function

Let $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $\frac{1}{2}+it$. 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.