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

On rr-gaps between zeros of the Riemann zeta-function

Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) > r + \Theta\sqrt{r}$ and $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) < r - \vartheta\sqrt{r}$ for explicit absolute positive constants $\Theta$ and $\vartheta$, where $\gamma$ denotes an ordinate of a zero of the Riemann zeta-function on the critical line. Selberg published announcements of this result several times but did not include a proof. We also suggest a general framework which might lead to stronger statements concerning the vertical distribution of nontrivial zeros of the Riemann zeta-function.Comment: to appear in the Bulletin of the London Mathematical Societ

Bagchi's Theorem for families of automorphic forms

We prove a version of Bagchi's Theorem and of Voronin's Universality Theorem for family of primitive cusp forms of weight $2$ and prime level, and discuss under which conditions the argument will apply to general reasonable family of automorphic $L$-functions.Comment: 15 page

Open circular billiards and the Riemann hypothesis

A comparison of escape rates from one and from two holes in an experimental container (e.g. a laser trap) can be used to obtain information about the dynamics inside the container. If this dynamics is simple enough one can hope to obtain exact formulas. Here we obtain exact formulas for escape from a circular billiard with one and with two holes. The corresponding quantities are expressed as sums over zeroes of the Riemann zeta function. Thus we demonstrate a direct connection between recent experiments and a major unsolved problem in mathematics, the Riemann hypothesis.Comment: 5 pages, 4 embedded postscript figures; v2: more explicit on how the Reimann Hypothesis arises from a comparison of one and two hole escape rate

Correlations of eigenvalues and Riemann zeros

We present a new approach to obtaining the lower order terms for $n$-correlation of the zeros of the Riemann zeta function. Our approach is based on the `ratios conjecture' of Conrey, Farmer, and Zirnbauer. Assuming the ratios conjecture we prove a formula which explicitly gives all of the lower order terms in any order correlation. Our method works equally well for random matrix theory and gives a new expression, which is structurally the same as that for the zeta function, for the $n$-correlation of eigenvalues of matrices from U(N)