12 research outputs found
From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond
The discovery of connections between the distribution of energy levels of
heavy nuclei and spacings between prime numbers has been one of the most
surprising and fruitful observations in the twentieth century. The connection
between the two areas was first observed through Montgomery's work on the pair
correlation of zeros of the Riemann zeta function. As its generalizations and
consequences have motivated much of the following work, and to this day remains
one of the most important outstanding conjectures in the field, it occupies a
central role in our discussion below. We describe some of the many techniques
and results from the past sixty years, especially the important roles played by
numerical and experimental investigations, that led to the discovery of the
connections and progress towards understanding the behaviors. In our survey of
these two areas, we describe the common mathematics that explains the
remarkable universality. We conclude with some thoughts on what might lie ahead
in the pair correlation of zeros of the zeta function, and other similar
quantities.Comment: Version 1.1, 50 pages, 6 figures. To appear in "Open Problems in
Mathematics", Editors John Nash and Michael Th. Rassias. arXiv admin note:
text overlap with arXiv:0909.491
A Random Matrix Model for Elliptic Curve L-Functions of Finite Conductor
We propose a random matrix model for families of elliptic curve L-functions
of finite conductor. A repulsion of the critical zeros of these L-functions
away from the center of the critical strip was observed numerically by S. J.
Miller in 2006; such behaviour deviates qualitatively from the conjectural
limiting distribution of the zeros (for large conductors this distribution is
expected to approach the one-level density of eigenvalues of orthogonal
matrices after appropriate rescaling).Our purpose here is to provide a random
matrix model for Miller's surprising discovery. We consider the family of even
quadratic twists of a given elliptic curve. The main ingredient in our model is
a calculation of the eigenvalue distribution of random orthogonal matrices
whose characteristic polynomials are larger than some given value at the
symmetry point in the spectra. We call this sub-ensemble of SO(2N) the excised
orthogonal ensemble. The sieving-off of matrices with small values of the
characteristic polynomial is akin to the discretization of the central values
of L-functions implied by the formula of Waldspurger and Kohnen-Zagier.The
cut-off scale appropriate to modeling elliptic curve L-functions is
exponentially small relative to the matrix size N. The one-level density of the
excised ensemble can be expressed in terms of that of the well-known Jacobi
ensemble, enabling the former to be explicitly calculated. It exhibits an
exponentially small (on the scale of the mean spacing) hard gap determined by
the cut-off value, followed by soft repulsion on a much larger scale. Neither
of these features is present in the one-level density of SO(2N). When N tends
to infinity we recover the limiting orthogonal behaviour. Our results agree
qualitatively with Miller's discrepancy. Choosing the cut-off appropriately
gives a model in good quantitative agreement with the number-theoretical data.Comment: 38 pages, version 2 (added some plots
THE DISTRIBUTION OF THE IRREDUCIBLES IN AN ALGEBRAIC NUMBER FIELD
We study the distribution of principal ideals generated by irreducible elements in an algebraic number field