A note on the gaps between consecutive zeros of the Riemann zeta-function

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

doi:10.1090/s0002-9939-2010-10443-4

Cited by 26 publications

(27 citation statements)

References 11 publications

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“…By a different method, also assuming RH, Feng and Wu [17] have proved that µ ≤ 0.5154. This improves previous estimates by a number of other authors [3,12,37].…”

Section: 43supporting

confidence: 87%

Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function

Carneiro

Chandee

Littmann

et al. 2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function N (T, β) defined to be the number of pairs γ and γ of ordinates of nontrivial zeros of the Riemann zetafunction satisfying 0 < γ, γ ≤ T and 0 < γ − γ ≤ 2πβ/ log T as T → ∞. In this paper, assuming the Riemann hypothesis, we prove upper and lower bounds for N (T, β), for all β > 0, using Montgomery's formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval [−β, β] in a way to minimize the L 1 R, 1 − sin πx πx 2 dx -error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work by P. X. Gallagher [19] in 1985, where the case β ∈ 1 2 N was considered using non-extremal majorants and minorants.

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“…By a different method, also assuming RH, Feng and Wu [17] have proved that µ ≤ 0.5154. This improves previous estimates by a number of other authors [3,12,37].…”

Section: 43supporting

confidence: 87%

Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function

Carneiro

Chandee

Littmann

et al. 2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…A different method of Mueller [25] has been used in a number of papers to prove lower bounds for λ conditional upon RH and its generalizations, see [7,11,13,14,16,24,26,27]. Our result in Theorem 1.1 that λ > 3.18 assuming RH supersedes all of these previous results.…”

Section: 2mentioning

confidence: 55%

Gaps between zeros of the Riemann zeta-function

Bui

Milinovich

2017

The Quarterly Journal of Mathematics

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We prove that there exist infinitely many consecutive zeros of the Riemann zetafunction on the critical line whose gaps are greater than 3.18 times the average spacing. Using a modification of our method, we also show that there are even larger gaps between the multiple zeros of the zeta function on the critical line (if such zeros exist).2010 Mathematics Subject Classification. 11M06, 11M26, 26D15.

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“…NN is supported in part by an NSERC Discovery grant. 1 In support of his conjecture, Gonek [5] has shown, assuming the Riemann Hypothesis and the simplicity of the zeros of ζ(s), that…”

Section: Introductionmentioning

confidence: 93%