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Large Gaps Between Zeros of the Zeta-Function on the Critical Line and Moment Conjectures from Random Matrix Theory

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Abstract

Denote by γn 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) 2kth and 4kth moment for the zeta-function originating from random matrix theory, we prove that for any fixed positive integer r more than cN(T) (log T)−4k 2 of the ordinates γn ∈ [0, T] satisfy

$$({\gamma_n+r}-\gamma_n) {{\rm log}\gamma_n \over 2\pi r} \geq \theta \ \ \ \ \ {\rm for \ any} \ \theta \leq {4k \over \pi er}$$

, where c is a computable positive constant depending on k, θ and r.

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Authors and Affiliations

  1. Departamento de Matemáticas, C. Universitaria de Cantoblanco, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Rasa Steuding

  2. Institut für Mathematik, Am Hubland, Universität Würzburg, 97074, Würzburg, Germany

    Jörn Steuding

Authors
  1. Rasa Steuding
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  2. Jörn Steuding
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Correspondence to Rasa Steuding.

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Dedicated to Professor Walter K. Hayman on the occasion of his 80th birthday

The authors were partially supported by Grant MTM2006-01859 of the Spanish MEC

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Steuding, R., Steuding, J. Large Gaps Between Zeros of the Zeta-Function on the Critical Line and Moment Conjectures from Random Matrix Theory. Comput. Methods Funct. Theory 8, 121–132 (2008). https://doi.org/10.1007/BF03321675

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  • DOI: https://doi.org/10.1007/BF03321675

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