Pair correlation estimates for the zeros of the zeta function via semidefinite programming

Chirre, Andrés; Gonçalves, Felipe; Laat, David de

doi:10.1016/j.aim.2019.106926

Cited by 15 publications

(20 citation statements)

References 42 publications

(78 reference statements)

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“…Regarding results that hold for a positive proportion of zeroes, Wu [Wu14] improved previous results by Conrey et al [CGG85] and Soundararajan [Sou96], that on RH one can take λ > 1.6989 and µ < 0.6553. Results holding for a positive proportion of zeroes under additional assumptions can be found in recent work of Chirre et al [CHdL20].…”

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confidence: 69%

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

Simonič¹,

Trudgian²,

Turnage-Butterbaugh³

2020

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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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confidence: 69%

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 proceed along these lines in the next subsection. We note that the larger class A has proved useful to sharpen some bounds in the theory of the Riemann zeta-function via sophisticated numerical experimentation [8] and, though numerics is not our main focus here, we have already laid the foundational theoretical framework for such endeavors.…”

Section: Strengths and Limitationsmentioning

confidence: 99%

On Montgomery's pair correlation conjecture: a tale of three integrals

Carneiro,

Chandee,

Chirre

et al. 2021

Preprint

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We study three integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. The first is the integral of Montgomery's function F pα, T q in bounded intervals, the second is an integral introduced by Selberg related to estimating the variance of primes in short intervals, and the last is the second moment of the logarithmic derivative of the Riemann zeta-function near the critical line. The conjectured asymptotic for any of these three integrals is equivalent to Montgomery's pair correlation conjecture. Assuming the Riemann hypothesis, we substantially improve the known upper and lower bounds for these integrals by introducing new connections to certain extremal problems in Fourier analysis. In an appendix, we study the intriguing problem of establishing the sharp form of an embedding between two Hilbert spaces of entire functions naturally connected to Montgomery's pair correlation conjecture.

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“…For the details, see [2,22,23]. The consequence of its truth implies that Lindelöf conjecture ( [17], p.34), Montgomery pair correlation conjecture [38,39] and the Berry-Keating conjecture in quantum chaos [40,41] are true and is also related to the Yang-Lee theorem [42,43]. For the more equivalences of this conjecture, see [44].…”

Section: Introductionmentioning

confidence: 99%

On the Riemann Hypothesis for the Zeta Function

Yang¹

2021

Preprint

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Pair correlation estimates for the zeros of the zeta function via semidefinite programming

Abstract: In this paper we study the distribution of the non-trivial zeros of the zeta-function ζ(s) (and other L-functions) under Montgomery's pair correlation approach. We use semidefinite programming to improve the asymptotic bounds for N * (T ), N d (T ) and N (λ, T ).

Cited by 15 publications

References 42 publications

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

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

On Montgomery's pair correlation conjecture: a tale of three integrals

On the Riemann Hypothesis for the Zeta Function

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