The distribution of spacings between quadratic residues

We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among other things, implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution.Comment: 38 pages; introduction and section 6.2 revised, references updated. To appear in Duke Math. Journa

Hecke theory and equidistribution for the quantization of linear maps of the torus

We study semi-classical limits of eigenfunctions of a quantized linear hyperbolic automorphism of the torus ("cat map"). For some values of Planck's constant, the spectrum of the quantized map has large degeneracies. Our first goal in this paper is to show that these degeneracies are coupled to the existence of quantum symmetries. There is a commutative group of unitary operators on the state-space which commute with the quantized map and therefore act on its eigenspaces. We call these "Hecke operators", in analogy with the setting of the modular surface. We call the eigenstates of both the quantized map and of all the Hecke operators "Hecke eigenfunctions". Our second goal is to study the semiclassical limit of the Hecke eigenfunctions. We will show that they become equidistributed with respect to Liouville measure, that is the expectation values of quantum observables in these eigenstates converge to the classical phase-space average of the observable.Comment: 37 pages. New title. Spelling mistake in bibliography corrected. To appear in Duke Math.

The fluctuations in the number of points on a hyperelliptic curve over a finite field

The number of points on a hyperelliptic curve over a field of $q$ elements may be expressed as $q+1+S$ where $S$ is a certain character sum. We study fluctuations of $S$ as the curve varies over a large family of hyperelliptic curves of genus $g$. For fixed genus and growing $q$, Katz and Sarnak showed that $S/\sqrt{q}$ is distributed as the trace of a random $2g\times 2g$ unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the the limiting distribution of $S$ is that of a sum of $q$ independent trinomial random variables taking the values $\pm 1$ with probabilities $1/2(1+q^{-1})$ and the value 0 with probability $1/(q+1)$. When both the genus and the finite field grow, we find that $S/\sqrt{q}$ has a standard Gaussian distribution.Comment: 10 pages. Final versio

Linear statistics for zeros of Riemann's zeta function

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

Electrostatic interactions between discrete helices of charge

We analytically examine the pair interaction for parallel, discrete helices of charge. Symmetry arguments allow for the energy to be decomposed into a sum of terms, each of which has an intuitive geometric interpretation. Truncated Fourier expansions for these terms allow for accurate modeling of both the axial and azimuthal terms in the interaction energy and these expressions are shown to be insensitive to the form of the interaction. The energy is evaluated numerically through application of an Ewald-like summation technique for the particular case of unscreened Coulomb interactions between the charges of the two helices. The mode structures and electrostatic energies of flexible helices are also studied. Consequences of the resulting energy expressions are considered for both F-actin and A-DNA aggregates

Is the Sunyaev-Zeldovich effect responsible for the observed steepening in the spectrum of the Coma radio halo ?

The spectrum of the radio halo in the Coma cluster is measured over almost two decades in frequency. The current radio data show a steepening of the spectrum at higher frequencies, which has implications for models of the radio halo origin. There is an on-going debate on the possibility that the observed steepening is not intrinsic to the emitted radiation, but is instead caused by the SZ effect. Recently, the Planck satellite measured the SZ signal and its spatial distribution in the Coma cluster allowing to test this hypothesis. Using the Planck results, we calculated the modification of the radio halo spectrum by the SZ effect in three different ways. With the first two methods we measured the SZ-decrement within the aperture radii used for flux measurements of the halo at the different frequencies. First we adopted the global compilation of data from Thierbach et al. and a reference aperture radius consistent with those used by the various authors. Second we used the available brightness profiles of the halo at different frequencies to derive the spectrum within two fixed apertures, and derived the SZ-decrement using these apertures. As a third method we used the quasi-linear correlation between the y and the radio-halo brightness at 330 MHz discovered by Planck to derive the modification of the radio spectrum by the SZ-decrement in a way that is almost independent of the adopted aperture radius. We found that the spectral modification induced by the SZ-decrement is 4-5 times smaller than that necessary to explain the observed steepening. Consequently a break or cut-off in the spectrum of the emitting electrons is necessary to explain current data. We also show that, if a steepening is absent from the emitted spectrum, future deep observations at 5 GHz with single dishes are expected to measure a halo flux in a 40 arcmin radius that would be 7-8 times higher than currently seen.Comment: 8 pages, 6 figures, accepted in Astronomy and Astrophysics (date of acceptance 19/08/2013

Observed Faraday Effects in Damped Lyman-Alpha Absorbers and Lyman Limit Systems: The Magnetised Environment of Galactic Building Blocks at Redshift=2

Protogalactic environments are typically identified using quasar absorption lines, and these galactic building blocks can manifest as Damped Lyman-Alpha Absorbers (DLAs) and Lyman Limit Systems (LLSs). We use radio observations of Faraday effects to test whether DLAs and LLSs host a magnetised medium, by combining DLA and LLS detections throughout the literature with 1.4 GHz polarization data from the NRAO VLA Sky Survey (NVSS). We obtain a control, a DLA, and a LLS sample consisting of 114, 19, and 27 lines-of-sight respectively - all of which are polarized at $\ge8\sigma$ to ensure Rician bias is negligible. Using a Bayesian framework, we are unable to detect either coherent or random magnetic fields in DLAs: the regular coherent magnetic fields within the DLAs must be $\le2.8$ $\mu$G, and the lack of depolarization is consistent with the weakly magnetised gas in DLAs being non-turbulent and quiescent. However, we find mild suggestive evidence that LLSs have coherent magnetic fields: after controlling for the redshift-distribution of our data, we find a 71.5% probability that LLSs have a higher RM than a control sample. We also find strong evidence that LLSs host random magnetic fields, with a 95.5% probability that LLS lines-of-sight have lower polarized fractions than a control sample. The regular coherent magnetic fields within the LLSs must be $\le2.4$ $\mu$G, and the magnetised gas must be highly turbulent with a typical scale on the order of $\approx5$-20 pc, which is similar to that of the Milky Way. This is consistent with the standard dynamo pedagogy, whereby magnetic fields in protogalaxies increase in coherence and strength as a function of cosmic time. Our results are consistent with a hierarchical galaxy formation scenario, with the DLAs, LLSs, and strong magnesium II (MgII) systems exploring three different stages of magnetic field evolution in galaxies.Comment: Submitted to Ap