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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)

- J. B. CONREY, N. C. SNAITH
- 2005

In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted L-functions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number theory, such as lower order terms in the zero statistics of… (More)

- J. Brian Conrey, J. P. Keating, Michael Rubinstein, Nina C. Snaith
- Experimental Mathematics
- 2006

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)

- J. B. CONREY
- 2008

Let T N,χ p,k (x) be the characteristic polynomial of the Hecke operator T p acting on the space of cusp forms S k (N, χ). We describe the factorization of T N,χ p,k (x) mod ℓ as k varies, and we explicitly calculate those factorizations for N = 1 and small ℓ. These factorizations are used to deduce the irreducibility of certain T 1,1 q,k (x) from the… (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)

- J. Brian Conrey
- 2003

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)

- J. B. Conrey, A. Ghosh
- 1998

In 1918 Hardy and Littlewood2]proved that Z T 1 j(1=2 + it)j 2 dt T log T and in 1926 Ingham 4] showed that Z T 1 j(1=2 + it)j 4 dt 1 2 2 T log 4 T : In general, it is conjectured that if k > 0, then there exists a c k > 0 such that Z T 1 j(1=2 + it)j 2k dt c k T log k 2 T : No value has been suggested for c k if k is diierent from 0,1, or 2. In this paper,… (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)

- J B Conrey, D W Farmer
- 1995

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)