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

Large Gaps Between Zeros of the Zeta-Function on the Critical Line and Moment Conjectures from... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © 2008 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321675
Publisher site
See Article on Publisher Site

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.

Journal

Computational Methods and Function TheorySpringer Journals

Published: May 22, 2007

References