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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)

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)