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

Goldston, D. A.; Gonek, S. M.

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

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Let $πS(t)$ 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.
5 pages