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131
Probability laws related to the Jacobi theta and Riemann zeta functions, and the Brownian excursions
 Bulletin (New series) of the American Mathematical Society
"... Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional ..."
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Cited by 55 (12 self)
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Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws. Contents
Characteristic polynomials of random matrices
 Communications in Mathematical Physics
"... We have discussed earlier the correlation functions of the random variables det(λ−X) in which X is a random matrix. In particular the moments of the distribution of these random variables are universal functions, when measured in the appropriate units of the level spacing. When the λ’s, instead of b ..."
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Cited by 55 (0 self)
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We have discussed earlier the correlation functions of the random variables det(λ−X) in which X is a random matrix. In particular the moments of the distribution of these random variables are universal functions, when measured in the appropriate units of the level spacing. When the λ’s, instead of belonging to the bulk of the spectrum, approach the edge, a crossover takes place to an Airy or to a Bessel problem, and we consider here these modified classes of universality. Furthermore, when an external matrix source is added to the probability distribution of X, various new phenomenons may occur and one can tune the spectrum of this source matrix to new critical points. Again there are remarkably simple formulae for arbitrary source matrices, which allow us to compute the moments of the characteristic polynomials in these cases as well.
Mean values of Lfunctions and symmetry
 Int. Math. Res. Notices
"... Abstract. Recently Katz and Sarnak introduced the idea of a symmetry group attached to a family of L–functions, and they gave strong evidence that the symmetry group governs many properties of the distribution of zeros of the L–functions. We consider the mean–values of the L–functions and the mollif ..."
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Cited by 41 (14 self)
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Abstract. Recently Katz and Sarnak introduced the idea of a symmetry group attached to a family of L–functions, and they gave strong evidence that the symmetry group governs many properties of the distribution of zeros of the L–functions. We consider the mean–values of the L–functions and the mollified mean–square of the L–functions and find evidence that these are also governed by the symmetry group. We use recent work of Keating and Snaith to give a complete description of these mean values. We find a connection to the Barnes–Vignéras Γ2–function and to a family of self–similar functions. 1.
Random matrices: Universality of the local eigenvalue statistics, submitted
"... Abstract. This is a continuation of our earlier paper [25] on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in [25] from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality ..."
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Cited by 40 (1 self)
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Abstract. This is a continuation of our earlier paper [25] on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in [25] from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov [23] for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors. 1.
On the frequency of vanishing of quadratic twists of modular Lfunctions
 in Number theory for the millennium, I (Urbana, IL, 2000), 301–315, A K Peters
, 2002
"... Abstract. We present theoretical and numerical evidence for a random matrix theoretical approach to a conjecture about vanishings of quadratic twists of certain Lfunctions. In this paper we 1 present some evidence that methods from random matrix theory can give insight into the frequency of vanishi ..."
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Cited by 38 (15 self)
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Abstract. We present theoretical and numerical evidence for a random matrix theoretical approach to a conjecture about vanishings of quadratic twists of certain Lfunctions. 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 Lfunctions. The central question is the following: given a holomorphic newform f with integral coefficients and associated Lfunction Lf(s), for how many fundamental discriminants d with d  ≤ x, does Lf(s, χd), the Lfunction twisted by the real, primitive, Dirichlet character associated with the discriminant d, vanish at the center of the critical strip to order at least 2? This question is of particular interest in the case that the Lfunction is associated with an elliptic curve, in light of the conjecture of Birch and SwinnertonDyer. This case corresponds to weight k = 2. We will focus on this case for most of the paper, though we do make some remarks about higher weights (see (26) and below). Suppose that E/Q is an elliptic curve with associated Lfunction (1) LE(s) = for ℜs> 1. Then, as a consequence of the TaniyamaShimura conjecture, recently solved by Wiles, Taylor, ([W], [TW]), and Breuil, Conrad, and Diamond, LE is entire and satisfies a functional equation n=1 a ∗ n n s
Nonvanishing of quadratic Dirichlet Lfunctions at
 Annals of Math. 152 (2000), 447  488 s = 1
"... The Generalized Riemann Hypothesis (GRH) states that all nontrivial zeros of Dirichlet Lfunctions lie on the line Re(s) = 1 2. Further, it is believed that there are no Qlinear ..."
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Cited by 36 (6 self)
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The Generalized Riemann Hypothesis (GRH) states that all nontrivial zeros of Dirichlet Lfunctions lie on the line Re(s) = 1 2. Further, it is believed that there are no Qlinear
Bertini theorems over finite fields
 Ann. of Math
"... Abstract. Let X be a smooth quasiprojective subscheme of P n of dimension m ≥ 0 over Fq. Then there exist homogeneous polynomials f over Fq for which the intersection of X and the hypersurface f = 0 is smooth. In fact, the set of such f has a positive density, equal to ζX(m + 1) −1, where ζX(s) = Z ..."
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Cited by 33 (8 self)
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Abstract. Let X be a smooth quasiprojective subscheme of P n of dimension m ≥ 0 over Fq. Then there exist homogeneous polynomials f over Fq for which the intersection of X and the hypersurface f = 0 is smooth. In fact, the set of such f has a positive density, equal to ζX(m + 1) −1, where ζX(s) = ZX(q −s) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture. 1.
Multiple Dirichlet series and moments of zeta and L–functions
 PROC. OF THE GAUSSDIRICHLET CONFERENCE, GÖTTINGEN 2005, CLAY MATH. PROC., AMS
, 2001
"... This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as ..."
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Cited by 32 (8 self)
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This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic Lseries. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of central values of Dirichlet Lseries. The methods utilized to derive this result are the convexity principle for functions of several complex variables combined with a knowledge of groups of functional equations for certain multiple Dirichlet series.
Lowlying zeros of families of elliptic curves
, 2006
"... There is a growing body of evidence giving strong evidence that zeros of families of Lfunctions follow distribution laws of eigenvalues of random matrices. This philosophy is known as the random matrix model or the KatzSarnak philosophy. The random matrix model makes predictions for the average di ..."
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Cited by 29 (2 self)
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There is a growing body of evidence giving strong evidence that zeros of families of Lfunctions follow distribution laws of eigenvalues of random matrices. This philosophy is known as the random matrix model or the KatzSarnak philosophy. The random matrix model makes predictions for the average distribution of zeros near the central point for families of Lfunctions. We study the lowlying zeros for families of elliptic curve Lfunctions. For these Lfunctions there is special arithmetic interest in any zeros at the central point (by the conjecture of Birch and SwinnertonDyer and the impressive partial results towards resolving the conjecture). We calculate the density of the lowlying zeros for various families of elliptic curves. Our main foci are the family of all elliptic curves and a large family with positive rank. A main challenge has been to obtain results with test functions that are concentrated close to the origin since the central point is a location of great interest. An application is an improvement on the upper bound of the average rank of the family of all elliptic curves. We show that there is an extra contribution to the density of the lowlying zeros from the family with positive rank (presumably from the “extra ” zero at the central point). 1