A note on S(t) and the zeros of the Riemann zeta‐function

By D. Goldston, Steven M. Gonek. 2005

81 citations recorded by the source corpus.

Let π S(t) denote the argument of the Riemann zeta‐function at the point 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/53d7217c2943506340a9f1fc44918c67c864549d

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

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Citations in the corpus

Partial sums of the Möbius function (2007, background): “Moreover, Goldston and Gonek’s estimate (see Theorem 1 of [3]) that for large t and 0 < h ≤ √ t one has |N(t + h)−N(t)− h 2π log t 2π | ≤ ( 12 + o(1)) log t/ log log t, readily shows that criteria (ii) and (iii) are also met.”
Moments of the Riemann zeta function (2006, methodology): “There is certainly some scope to improve our Corollary C below, and it may be instructive to understand what the limit of the method would be (analogously to the elegant treatment of Im log ζ( 1 2 + it) given by Goldston and Gonek [5]).”
Bounding ζ(s) in the critical strip (2010, methodology): “The method of Goldston and Gonek [6] uses the explicit formula together with the classical Beurling-Selberg majorants and minorants of characteristic functions of intervals, and leads to the bound |S(t)| ≤ ( 1 2 + o(1) ) log t log log t .”
Bounding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(t)$$\end{document} and \documentclass[12pt]{minimal} \usepa (2012, methodology): “This is an important observation since Goldston and Gonek showed how to use (4) to bound the maximum multiplicity of a zero of ζ ( s ) and how to bound the maximum gap between consecutive zeros of the Riemann zeta-function on the critical line (see [12, Corollary 1]).”