Computational Methods and Function Theory 8 (2008), No. 1, 121--132
Copyright Heldermann Verlag 2008
Rasa Steuding
rasa.steuding@uam.es , Universidad Autónoma de Madrid, Departamento de Matemáticas, C. Universitaria de Cantoblanco, 28049 Madrid, Spain.
Jörn Steuding
steuding@mathematik.uni-wuerzburg.de , Universität Würzburg, Institut für Mathematik, Am Hubland, 97074 Würzburg, Germany.
Denote by $\gamma_n$ the positive ordinates of the non-trivial zeros of the zeta-function in ascending order. Assuming the Riemann hypothesis and conjectural asymptotic formulae for the (continuous and discrete) $2k$th and $4k$th moment for the zeta-function originating from random matrix theory, we prove that for any fixed positive integer $r$ more than $c N(T)(\log T)^{-4k^2}$ of the ordinates $\gamma_n\in[0,T]$ satisfy $$ (\gamma_{n+r}-\gamma_n)\frac{\log\gamma_n}{2\pi r}\geq\theta \qquad\mbox{for any }\theta\leq \frac{4k}{\pi er}, $$ where $c$ is a computable positive constant depending on $k$, $\theta$ and $r$.
Keywords: Riemann zeta-function, nontrivial zeros, Riemann hypothesis, pair correlation, spacing between consecutive zeros, random matrix theory.
MSC 2000: 11M06, 11M26.
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