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

D. A. Goldston; S. M. Gonek

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

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arxiv math/0511092 v1 pith:2UCXPM7C submitted 2005-11-03 math.NT

D. A. Goldston , S. M. Gonek This is my paper · ORCID

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abstract

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.

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Spectral Riccati--Gamma Concavity, Symmetric Zero Cancellation, and Conditional Criteria for the Riemann Hypothesis
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Rules out naive concavity criterion for RH and derives conditional zero-density and localisation criteria via a finite spectral Riccati-Gamma averaging framework.

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