Gaps between zeros of the Riemann zeta-function

Bui, H. M.; Milinovich, Micah B.

doi:10.1093/qmath/hax047

Cited by 13 publications

(18 citation statements)

References 38 publications

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“…Motivations for the extreme eigenvalues gaps statistics include relaxation time for diagonalization algorithms [2,12], conjectures in analytic number theory (e.g. the extreme gaps between zeros of the Riemann zeta function [2,10]), conjectures in algorithmic number theory (the Poisson ansatz for large gaps suggests the complexity of an algorithm to detect square free numbers [4]), and quantum chaos in the complementary Poissonian regime [3].…”

Section: Introductionmentioning

confidence: 99%

Extreme gaps between eigenvalues of Wigner matrices

Bourgade¹

2018

Preprint

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This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the entries.The proof relies on the Erdős-Schlein-Yau dynamic approach. We exhibit a new observable that satisfies a stochastic advection equation and reduces local relaxation of the Dyson Brownian motion to a maximum principle. This observable also provides a simple and unified proof of universality in the bulk and at the edge, which is quantitative. To illustrate this, we give the first explicit rate of convergence to the Tracy-Widom distribution for generalized Wigner matrices.

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Section: Introductionmentioning

confidence: 99%

Extreme gaps between eigenvalues of Wigner matrices

Bourgade¹

2018

Preprint

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“…Since the smallest value of β is α = β = 1/2, we see that we only recover (2) with this choice and do worse otherwise. 1 Now consider β < α. Thus β < 1/2, and we substitute α = 1/(4β).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…We refer the reader to [4,8] for the history of this problem. The best current results under RH are: µ ≤ 0.515396 by Preobrazhenski ȋ [7] and λ ≥ 3.18 by Bui and Milinovich [1]. The result of [7] is based on a method introduced by Montgomery and Odlyzko [6].…”

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confidence: 99%

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A limitation on proving the existence of small gaps between zeta-zeros

Goldston¹,

Trudgian²,

Turnage-Butterbaugh³

2022

Preprint

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We assume the Riemann Hypothesis (RH) in this paper. The existence of Landau-Siegel zeros (or the Alternative Hypothesis) implies that there are long ranges where the zeros of the Riemann zeta-function are always spaced no closer than one half of the average spacing. However, numerical evidence strongly agrees with the GUE model where there are a positive proportion of consecutive zeros within any small multiple of the average spacing. Currently, assuming RH, the best result known produces infinitely many consecutive zeros within 0.515396 times the average spacing. This is obtained using the Montgomery-Odlyzko (M-O) method. It is also known that the M-O method fails to prove the existence of consecutive zeros closer than 1/2 times the average spacing. It is a tantalizing hope that the M-O method could still obtain infinitely many consecutive zeros arbitrarily close to 1/2 times the average spacing. We prove however that the M-O method can never find infinitely many consecutive zeros within 0.5042 times the average spacing.

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“…Selberg's and Fujii's observations imply µ 0 < 1 < λ 0 . On the Riemann Hypothesis (RH), Preobrazhenski ȋ [Pre16] proved that µ 0 ≤ 0.515396; Bui-Milinovich [BM18] proved that on RH λ 0 ≥ 3.18. These are the current best results that hold for infinitely many pairs of zeroes under RH.…”

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confidence: 99%

Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function

Simonič¹,

Trudgian²,

Turnage-Butterbaugh³

2020

Preprint

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We make explicit an argument of Heath-Brown concerning large and small gaps between nontrivial zeroes of the Riemann zeta-function, ζ(s).In particular, we provide the first unconditional results on gaps (large and small) which hold for a positive proportion of zeroes. To do this we prove explicit bounds on the second and fourth power moments of S(t + h) − S(t), where S(t) denotes the argument of ζ(s) on the critical line and h ≪ 1/ log T . We also use these moments to prove explicit results on the density of the nontrivial zeroes of ζ(s) of a given multiplicity.

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Gaps between zeros of the Riemann zeta-function

Cited by 13 publications

References 38 publications

Extreme gaps between eigenvalues of Wigner matrices

Extreme gaps between eigenvalues of Wigner matrices

A limitation on proving the existence of small gaps between zeta-zeros

Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function

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