Triple correlation of the Riemann zeros

We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested in 1996 by Bogomolny and Keating taking inspiration from semi-classical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously if one assumes the ratios conjecture of Conrey, Farmer and Zirnbauer. Bogomolny and Keating returned to their previous results simultaneously with this current work, and have written out the full expression. The result presented in this paper agrees precisely with their formula, as well as with our numerical computations, which we include here. We also include an alternate proof of the triple correlation of eigenvalues from random U(N) matrices which follows a nearly identical method to that for the Riemann zeros, but is based on the theorem for averages of ratios of characteristic polynomials

Riemann Zeros and Random Matrix Theory

In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much recent work concerning other varieties of L-functions, this article will concentrate on the zeta function as the simplest example illustrating the role of random matrix theory.

Developments in Random Matrix Theory

In this preface to the Journal of Physics A, Special Edition on Random Matrix Theory, we give a review of the main historical developments of random matrix theory. A short summary of the papers that appear in this special edition is also given.Comment: 22 pages, Late

Discretisation for odd quadratic twists

The discretisation problem for even quadratic twists is almost understood, with the main question now being how the arithmetic Delaunay heuristic interacts with the analytic random matrix theory prediction. The situation for odd quadratic twists is much more mysterious, as the height of a point enters the picture, which does not necessarily take integral values (as does the order of the Shafarevich-Tate group). We discuss a couple of models and present data on this question.Comment: To appear in the Proceedings of the INI Workshop on Random Matrix Theory and Elliptic Curve

Random Matrix Theory and the Fourier Coefficients of Half-Integral Weight Forms

Conjectured links between the distribution of values taken by the characteristic polynomials of random orthogonal matrices and that for certain families of L-functions at the centre of the critical strip are used to motivate a series of conjectures concerning the value-distribution of the Fourier coefficients of half-integral weight modular forms related to these L-functions. Our conjectures may be viewed as being analogous to the Sato-Tate conjecture for integral weight modular forms. Numerical evidence is presented in support of them.Comment: 28 pages, 8 figure

Stochastic Models for Replication Origin Spacings in Eukaryotic DNA Replication

We consider eukaryotic DNA replication and in particular the role of replication origins in this process. We focus on origins which are `active' - that is, trigger themselves in the process before being read by the replication forks of other origins. We initially consider the spacings of these active replication origins in comparison to certain probability distributions of spacings taken from random matrix theory. We see how the spacings between neighbouring eigenvalues from certain collections of random matrices has some potential for modelling the spacing between active origins. This suitability can be further augmented with the use of uniform thinning which acts as a continuous deformation between correlated eigenvalue spacings and exponential (Poissonian) spacings. We model the process as a modified 2D Poisson process with an added exclusion rule to identify active points based on their position on the chromosome and trigger time relative to other origins. We see how this can be reduced to a stochastic geometry problem and show analytically that two active origins are unlikely to be close together, regardless of how many non-active points are between them. In particular, we see how these active origins repel linearly. We then see how data from various DNA datasets match with simulations from our model. We see that whilst there is variety in the DNA data, comparing the data with the model provides insight into the replication origin distribution of various organisms.Comment: 18 pages, 26 figure