C. P. Hughes

Random matrix theory and the derivative of the Riemann zeta function

Random matrix theory is used to model the asymptotics of the discrete moments of the derivative of the Riemann zeta function, ζ(s), evaluated at the complex zeros ½; + iγn. We also discuss the probability distribution of ln |ζ′(1/2 + iγn)|, proving the central limit theorem for the corresponding random matrix distribution and analysing its large deviations.

The zeros of random polynomials cluster uniformly near the unit circle

In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erd˝ os and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomial.

The maximum size of L-functions

Abstract We conjecture the true rate of growth of the maximum size of the Riemann zeta-function and other L-functions. We support our conjecture using arguments from random matrix theory, conjectures for moments of L-functions, and also by assuming a random model for the primes.

A hybrid Euler-Hadamard product for the Riemann zeta function

We use a smoothed version of the explicit formula to find an accurate pointwise approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function which involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. For the second and fourth moments, we establish all of the steps in our approach rigorously. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory

Low-lying zeros of

We investigate the moments of a smooth counting function of the zeros near the central point of L-functions of weight k cuspidal newforms of prime level N. We split by the sign of the functional equations and show that for test functions whose Fourier transform is supported in (-1/n, 1/n), as N --> oo the first n centered moments are Gaussian. By extending the support to (-1/n-1, 1/n-1), we see non-Gaussian behavior; in particular the odd centered moments are non-zero for such test functions. If we do not split by sign, we obtain Gaussian behavior for support in (-2/n, 2/n) if 2k >= n. The nth centered moments agree with Random Matrix Theory in this extended range, providing additional support for the Katz-Sarnak conjectures. The proof requires calculating multidimensional integrals of the non-diagonal terms in the Bessel-Kloosterman expansion of the Petersson formula. We convert these multidimensional integrals to one-dimensional integrals already considered in the work of Iwaniec-Luo-Sarnak, and derive a new and more tractable expression for the nth centered moments for such test functions. This new formula facilitates comparisons between number theory and random matrix theory for test functions supported in (-1/n-1, 1/n-1) by simplifying the combinatorial arguments. As an application we obtain bounds for the percentage of such cusp forms with a given order of vanishing at the central point.

L

We consider a smooth counting function of the scaled zeros of the Riemann zeta function, around height T. We show that the first few moments tend to the Gaussian moments, with the exact number depending on the statistic considered. To cite this article: C.P. Hughes, Z. Rudnick, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 667–670.

-functions with orthogonal symmetry

In this paper, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin Fourier transform of such a random polynomial, first obtained by Keating and Snaith in (7), using a simple recursion formula, and from there we are able to obtain the joint law of its radial and an- gular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of in- dependent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith in (7) is now obtained from the classical central limit theorems of Probability Theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm type results.

Linear statistics for zeros of Riemann's zeta function

We consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups. We compute their moments and find that the first few moments are Gaussian, whereas the limiting distribution is not. The precise number of Gaussian moments depends upon the particular statistic considered.

The characteristic polynomial of a random unitary matrix: A probabilistic approach

We calculate the discrete moments of the characteristic polynomial of a random unitary matrix, evaluated a small distance away from an eigenangle. Such results allow us to make conjectures about similar moments for the Riemann zeta function, and provide a uniform approach to understanding moments of the zeta function and its derivative.

Mock-Gaussian behaviour for linear statistics of classical compact groups

Abstract We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length .