A Central Limit Theorem for the Zeroes of the Zeta Function

Rodgers, Brad

doi:10.1142/s1793042113501054

Cited by 14 publications

(24 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It turns out that the appearance of the H 1/2 -Gaussian noise is remarkably universal in the mesoscopic limit of one dimensional ensembles with random matrix type repulsion and this problem has attracted renewed interest in the last couple of years. For instance, the analogue of Soshnikov's CLT (1.9) was obtained for the GUE [28], more general invariant ensembles [12,46], Wigner matrices [23,47,34], β-ensembles [10,2] and for zeros of the Riemann zeta function [11,63]. It is likely the counterpart of Theorem 1.3 continues to hold for these models as well.…”

Section: Background and Results For The Cuementioning

confidence: 90%

Subcritical Multiplicative Chaos for Regularized Counting Statistics from Random Matrix Theory

Lambert

Ostrovsky

Simm

2018

Commun. Math. Phys.

View full text Add to dashboard Cite

For an N × N Haar distributed random unitary matrix UN , we consider the random field defined by counting the number of eigenvalues of UN in a mesoscopic arc centered at the point u on the unit circle. We prove that after regularizing at a small scale ǫN > 0, the renormalized exponential of this field converges as N → ∞ to a Gaussian multiplicative chaos measure in the whole subcritical phase. We discuss implications of this result for obtaining a lower bound on the maximum of the field. We also show that the moments of the total mass converge to a Selberg-like integral and by taking a further limit as the size of the arc diverges, we establish part of the conjectures in [55]. By an analogous construction, we prove that the multiplicative chaos measure coming from the sine process has the same distribution, which strongly suggests that this limiting object should be universal. Our approach to the L 1 -phase is based on a generalization of the construction in Berestycki [5] to random fields which are only asymptotically Gaussian. In particular, our method could have applications to other random fields coming from either random matrix theory or a different context.

show abstract

Section: Background and Results For The Cuementioning

confidence: 90%

Subcritical Multiplicative Chaos for Regularized Counting Statistics from Random Matrix Theory

Lambert

Ostrovsky

Simm

2018

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…We begin this section with a brief description of the statistic of Riemann zeroes that was introduced by Bourgade and Kuan [15] and Rodgers [53] (henceforth referred to as the BKR statistic), following the approach and notations of Bourgade and Kuan. The Riemann zeta function is defined 4…”

Section: Resultsmentioning

confidence: 99%

“…In this paper we contribute to the literature on the statistical distribution of Riemann zeroes in the mesoscopic regime. The study of the values of the Riemann zeta function in the mesoscopic regime was pioneered by Selberg [57], [58] and then extended to the distribution of zeroes by Fujii [23], Hughes and Rudnick [32], Bourgade [14], and most recently by Bourgade and Kuan [15], Rodgers [53], and Kargin [35]. Assuming the Riemann hypothesis (except for Selberg), these authors rigorously established various central limit theorems for the distribution of Riemann zeroes.…”

Section: Introductionmentioning

confidence: 99%

On Riemann zeroes, lognormal multiplicative chaos, and Selberg integral

Ostrovsky¹

2016

Nonlinearity

View full text Add to dashboard Cite

Rescaled Mellin-type transforms of the exponential functional of the Bourgade-Kuan-Rodgers statistic of Riemann zeroes are conjecturally related to the distribution of the total mass of the limit lognormal stochastic measure of Mandelbrot-Bacry-Muzy. The conjecture implies that a non-trivial, log-infinitely divisible probability distribution is associated with Riemann zeroes. For application, integral moments, covariance structure, multiscaling spectrum, and asymptotics associated with the exponential functional are computed in closed form using the known meromorphic extension of the Selberg integral.

show abstract

“…Within other symmetry classes and for the Dyson's β-ensembles, mesoscopic correlations are also conjectured to be described by the H 1/2 -Gaussian field. For instance, this has been rigorously established for the Gaussian β-Ensembles in [4], for random matrices from the special orthogonal and symplectic groups in [48] and in number theory, when considering mesoscopic linear statistics of the zeros of the Riemann-Zeta function [5,46]. There are also examples of determinantal processes where the precise asymptotics of the correlation kernel is not known, but the CLT (1.15) has been proved by other means, e.g.…”

Section: )mentioning

confidence: 99%

Mesoscopic fluctuations for unitary invariant ensembles

Lambert¹

2018

Electron. J. Probab.

View full text Add to dashboard Cite

Considering a determinantal point process on the real line, we establish a connection between the sine-kernel asymptotics for the correlation kernel and the CLT for mesoscopic linear statistics. This implies universality of mesoscopic fluctuations for a large class of unitary invariant Hermitian ensembles. In particular, this shows that the support of the equilibrium measure need not be connected in order to see Gaussian fluctuations at mesoscopic scales. Our proof is based on the cumulants computations introduced in [48] for the CUE and the sine process and the asymptotics formulae derived by Deift et al. [12]. For varying weights e −N Tr V (H) , in the one-cut regime, we also provide estimates for the variance of linear statistics Tr f (H) which are valid for a rather general function f . In particular, this implies that the characteristic polynomials of such Hermitian random matrices converge in a suitable regime to a regularized fractional Brownian motion with logarithmic correlations introduced in [19]. For the GUE kernel, we also discuss how to obtain the necessary sine-kernel asymptotics at mesoscopic scale by elementary means.

show abstract

A Central Limit Theorem for the Zeroes of the Zeta Function

Cited by 14 publications

References 34 publications

Subcritical Multiplicative Chaos for Regularized Counting Statistics from Random Matrix Theory

Subcritical Multiplicative Chaos for Regularized Counting Statistics from Random Matrix Theory

On Riemann zeroes, lognormal multiplicative chaos, and Selberg integral

Mesoscopic fluctuations for unitary invariant ensembles

Contact Info

Product

Resources

About