Twin prime correlations from the pair correlation of Riemann zeros

We establish, via a formal/heuristic Fourier inversion calculation, that the Hardy-Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height $E$ on the critical line. Previously it was known that the Hardy-Littlewood conjecture implies the pair correlation formula, and we show that the reverse implication also holds. A smooth form of the Hardy-Littlewood conjecture is obtained by inverting the $E \rightarrow \infty$ limit of the two-point correlation function and the precise form of the conjecture is found by including asymptotically lower order terms in the two-point correlation function formula.Comment: 11 page

On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class

We establish the equivalence of conjectures concerning the pair correlation of zeros of $L$-functions in the Selberg class and the variances of sums of a related class of arithmetic functions over primes in short intervals. This extends the results of Goldston & Montgomery [7] and Montgomery & Soundararajan [11] for the Riemann zeta-function to other $L$-functions in the Selberg class. Our approach is based on the statistics of the zeros because the analogue of the Hardy-Littlewood conjecture for the auto-correlation of the arithmetic functions we consider is not available in general. One of our main findings is that the variances of sums of these arithmetic functions over primes in short intervals have a different form when the degree of the associated $L$-functions is 2 or higher to that which holds when the degree is 1 (e.g. the Riemann zeta-function). Specifically, when the degree is 2 or higher there are two regimes in which the variances take qualitatively different forms, whilst in the degree-1 case there is a single regime

Quantum chaotic resonances from short periodic orbits

We present an approach to calculating the quantum resonances and resonance wave functions of chaotic scattering systems, based on the construction of states localized on classical periodic orbits and adapted to the dynamics. Typically only a few of such states are necessary for constructing a resonance. Using only short orbits (with periods up to the Ehrenfest time), we obtain approximations to the longest living states, avoiding computation of the background of short living states. This makes our approach considerably more efficient than previous ones. The number of long lived states produced within our formulation is in agreement with the fractal Weyl law conjectured recently in this setting. We confirm the accuracy of the approximations using the open quantum baker map as an example.Comment: 4 pages, 4 figure

Autocorrelation of Random Matrix Polynomials

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions

Periodic orbit bifurcations and scattering time delay fluctuations

We study fluctuations of the Wigner time delay for open (scattering) systems which exhibit mixed dynamics in the classical limit. It is shown that in the semiclassical limit the time delay fluctuations have a distribution that differs markedly from those which describe fully chaotic (or strongly disordered) systems: their moments have a power law dependence on a semiclassical parameter, with exponents that are rational fractions. These exponents are obtained from bifurcating periodic orbits trapped in the system. They are universal in situations where sufficiently long orbits contribute. We illustrate the influence of bifurcations on the time delay numerically using an open quantum map.Comment: 9 pages, 3 figures, contribution to QMC200

Number fields and function fields:Coalescences, contrasts and emerging applications

The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. We here introduce the papers published in this Theo Murphy meeting issue, where some of the recent developments are explained

Quantum statistics on graphs

Quantum graphs are commonly used as models of complex quantum systems, for example molecules, networks of wires, and states of condensed matter. We consider quantum statistics for indistinguishable spinless particles on a graph, concentrating on the simplest case of abelian statistics for two particles. In spite of the fact that graphs are locally one-dimensional, anyon statistics emerge in a generalized form. A given graph may support a family of independent anyon phases associated with topologically inequivalent exchange processes. In addition, for sufficiently complex graphs, there appear new discrete-valued phases. Our analysis is simplified by considering combinatorial rather than metric graphs -- equivalently, a many-particle tight-binding model. The results demonstrate that graphs provide an arena in which to study new manifestations of quantum statistics. Possible applications include topological quantum computing, topological insulators, the fractional quantum Hall effect, superconductivity and molecular physics.Comment: 21 pages, 6 figure

Quantization of multidimensional cat maps

In this work we study cat maps with many degrees of freedom. Classical cat maps are classified using the Cayley parametrization of symplectic matrices and the closely associated center and chord generating functions. Particular attention is dedicated to loxodromic behavior, which is a new feature of two-dimensional maps. The maps are then quantized using a recently developed Weyl representation on the torus and the general condition on the Floquet angles is derived for a particular map to be quantizable. The semiclassical approximation is exact, regardless of the dimensionality or of the nature of the fixed points.Comment: 33 pages, latex, 6 figures, Submitted to Nonlinearit

Entanglement in Quantum Spin Chains, Symmetry Classes of Random Matrices, and Conformal Field Theory

We compute the entropy of entanglement between the first $N$ spins and the rest of the system in the ground states of a general class of quantum spin-chains. We show that under certain conditions the entropy can be expressed in terms of averages over ensembles of random matrices. These averages can be evaluated, allowing us to prove that at critical points the entropy grows like $\kappa\log_2 N + {\tilde \kappa}$ as $N\to\infty$, where $\kappa$ and ${\tilde \kappa}$ are determined explicitly. In an important class of systems, $\kappa$ is equal to one-third of the central charge of an associated Virasoro algebra. Our expression for $\kappa$ therefore provides an explicit formula for the central charge.Comment: 4 page

Trajectories of educational aspirations through high school and beyond: A gendered phenomenon?

Growth curve modeling was utilized to examine change in the educational aspirations of adolescents from early high school through to three years beyond high school, as a function of gender and other adolescent characteristics. Significant gender effects were found for level of education aspired to, rate of growth, and degree of acceleration: boys’ aspirations were lower in early high school, accelerated at a faster pace to peak above girls’ aspirations by the end of high school, and dropped more steeply so that, by the post-high school period, educational aspirations were equivalent across genders. Gender also interacted with Grade 9 achievement in determining educational trajectories. Finally, the perception that one faces barriers in educational attainment was found to significantly influence rate of growth and acceleration, indicating that change in aspirations over time differs between people who see barriers to their education and people who do not, independently of gender. Implications of these results for promoting students’ educational aspirations are discussed.Keywords: Gender, high school, educational aspirationsLa modélisation des courbes de croissance a été utilisée pour examiner les changements dans les ambitions scolaires des adolescents à l'école secondaire et jusqu'à trois ans au-delà de l'école secondaire, en fonction du sexe et d'autres caractéristiques des adolescents. D'importantes différences selon le sexe ont été observées concernant le niveau d'éducation auquel les adolescents aspiraient, le taux de croissance, et le degré d'accélération: la courbe représentant les ambitions des garçons était plus basse au début de l'école secondaire, augmentait ensuite à un rythme plus rapide jusqu'à atteindre un pic au-dessus de la courbe des filles pour la période de la fin de l'école secondaire, et chutait enfin plus fortement de telle sorte que les deux courbes se retrouvaient au même niveau dans la période post-secondaire. Le genre a également interagi avec la réussite en neuvième année en déterminant des trajectoires éducatives. Enfin, la perception d'être confronté à des barrières entravant la réussite scolaire s'est révélée être un facteur qui influence de manière significative le taux de croissance et d'accélération, ce qui indique que le changement dans les ambitions scolaires au fil du temps diffère entre ceux qui perçoivent des obstacles au cours de leur scolarité et ceux qui n'en perçoivent pas, indépendamment de leur sexe. Les implications de ces résultats pour promouvoir les ambitions scolaires des élèves sont discutées.Mots-clés: Genre, école secondaire, aspirations scolaire