J. Brian Conrey

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We present theoretical and numerical evidence for a random matrix theoretical approach to a conjecture about vanishings of quadratic twists of certain L-functions. In this paper we 1 present some evidence that methods from random matrix theory can give insight into the frequency of vanishing for quadratic twists of modular L-functions. The central question(More)
Conjectured links between the distribution of values taken by the characteristic polynomials of random orthogonal matrices and that for certain families of L-functions at the centre of the critical strip are used to motivate a series of conjectures concerning the value-distribution of the Fourier coefficients of half-integral weight modular forms related to(More)
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 U Sp(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,(More)
H ilbert, in his 1900 address to the Paris International Congress of Mathematicians , listed the Riemann Hypothesis as one of his 23 problems for mathematicians of the twentieth century to work on. Now we find it is up to twenty-first century mathematicians! The Riemann Hypothesis (RH) has been around for more than 140 years, and yet now is arguably the(More)
— For the classical compact Lie groups K ≡ U N the autocorrelation functions of ratios of characteristic polynomials (z , w) → Det(z − k)/Det(w − k) are studied with k ∈ K as random variable. Basic to our treatment is a property shared by the spinor representation of the spin group with the Shale-Weil representation of the metaplectic group: in both cases(More)
Associated to a newform f (z) is a Dirichlet series L f (s) with functional equation and Euler product. Hecke showed that if the Dirichlet series F (s) has a functional equation of the appropriate form, then F (s) = L f (s) for some holomorphic newform f (z) on ?(1). Weil extended this result to ? 0 (N) under an assumption on the twists of F (s) by(More)