Michael O. Rubinstein

Integral moments of L -functions

We give a new heuristic for all of the main terms in the integral moments of various families of primitive

Chebyshev's bias

L

LOW-LYING ZEROS OF L-FUNCTIONS AND RANDOM MATRIX THEORY

-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical

Autocorrelation of random matrix polynomials

L

The Number of Intersection Points Made by the Diagonals of a Regular Polygon

-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

On a conjecture of Helleseth regarding pairs of binary m-sequences

The title refers to the fact, noted by Chebyshev in 1853, that primes congruent to 3 modulo 4 seem to predominate over those congruent to 1. We study this phenomenon and its generalizations. Assuming the Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis (about the zeros of the Dirichlet L-function), we can characterize exactly those moduli and residue classes for which the bias is present. We also give results of numerical investigations on the prevalence of the bias for several moduli. Finally, we briefly discuss generalizations of the bias to the distribution to primes in ideal classes in number fields, and to prime geodesics in homology classes on hyperbolic surfaces.

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

By looking at the average behavior (n-level density) of the low-lying zeros of certain families of L-functions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups.

Ranks of Elliptic Curves and Random Matrix Theory: Secondary terms in the number of vanishings of quadratic twists of elliptic curve L -functions

Abstract:u2002We 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 the 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.

A simple heuristic proof of Hardy and Littlewood's conjecture B

We give a formula for the number of interior intersection points made by the diagonals of a regular n-gon. The answer is a polynomial on each residue class modulo 2520. We also compute the number of regions formed by the diagonals, by using Eulers formula V - E + F = 2.

Conjectures and Experiments Concerning the Moments of L(1/2, χ d )

Binary m-sequences are maximal-length sequences generated by shift registers of length m, that are employed in navigation, radar, and spread-spectrum communication. It is well known that given a pair of distinct m-sequences, the crosscorrelation function must take on at least three values. This correspondence addresses a conjecture made by Helleseth in 1976, that if m is a power of 2, then there are no pairs of binary m-sequences with a 3-valued crosscorrelation function. This conjecture is proved under the assumption that the three correlation values are symmetric about -1.