Results 1  10
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30
Zeroes of Zeta Functions and Symmetry
, 1999
"... Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of cur ..."
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Cited by 105 (2 self)
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Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the lowlying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and Lfunctions.
Computational Strategies for the Riemann Zeta Function
 Journal of Computational and Applied Mathematics
, 2000
"... We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of th ..."
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Cited by 46 (9 self)
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We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call "value recycling".
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 40 (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.
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
Autocorrelation of random matrix polynomials
 COMMUN. MATH. PHYS
, 2003
"... 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 USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in t ..."
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Cited by 32 (17 self)
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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 USp(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, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than largematrix asymptotic approximations. They also mirror exactly the autocorrelation formulae conjectured to hold for Lfunctions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of Lfunctions.
Lowlying zeros of Lfunctions and random matrix theory
 Duke Math. J
, 2001
"... By looking at the average behavior (nlevel density) of the lowlying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. 1. ..."
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Cited by 31 (0 self)
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By looking at the average behavior (nlevel density) of the lowlying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. 1.
Random matrices and Lfunctions
 J. PHYS A MATH GEN
, 2003
"... In recent years there has been a growing interest in connections between the statistical properties of number theoretical Lfunctions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications. ..."
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Cited by 19 (7 self)
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In recent years there has been a growing interest in connections between the statistical properties of number theoretical Lfunctions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications.
Investigations of zeros near the central point of elliptic curve Lfunctions
"... We explore the effect of zeros at the central point on nearby zeros of elliptic curve Lfunctions, especially for oneparameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman’s Specialization Theorem, for t sufficiently large the Lfunction of each curve Et in t ..."
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Cited by 16 (4 self)
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We explore the effect of zeros at the central point on nearby zeros of elliptic curve Lfunctions, especially for oneparameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman’s Specialization Theorem, for t sufficiently large the Lfunction of each curve Et in the family has r zeros (called the family zeros) at the central point. We observe experimentally a repulsion of the zeros near the central point, and the repulsion increases with r. There is greater repulsion in the subset of curves of rank r + 2 than in the subset of curves of rank r in a rank r family. For curves with comparable conductors, the behavior of rank 2 curves in a rank 0 oneparameter family over Q is statistically different from that of rank 2 curves from a rank 2 family. Unlike excess rank calculations, the repulsion decreases markedly as the conductors increase, and we conjecture that the r family zeros do not repel in the limit. Finally, the differences between adjacent normalized zeros near the central point are statistically independent of the repulsion, family rank and rank of the curves in the subset. Specifically, the normalized differences are statistically equal for all curves investigated with rank 0, 2 or 4 and comparable conductors from oneparameter families of rank 0 or 2 over Q. 1
Derivatives of random matrix characteristic polynomials with applications to elliptic curves
 J. Phys. A
"... The value distribution of derivatives of characteristic polynomials of matrices from SO(N) is calculated at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. We consider subsets of matrices from SO(N) that are constrained to have at least n eigenvalues equal to ..."
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Cited by 13 (3 self)
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The value distribution of derivatives of characteristic polynomials of matrices from SO(N) is calculated at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. We consider subsets of matrices from SO(N) that are constrained to have at least n eigenvalues equal to 1, and investigate the first nonzero derivative of the characteristic polynomial at that point. The connection between the values of random matrix characteristic polynomials and values of Lfunctions in families has been wellestablished. The motivation for this work is the expectation that through this connection with Lfunctions derived from families of elliptic curves, and using the Birch and SwinnertonDyer conjecture to relate values of the Lfunctions to the rank of elliptic curves, random matrix theory will be useful in probing important questions concerning these ranks. 1
INTEGRAL MOMENTS OF LFUNCTIONS
, 2005
"... We give a newheuristic for all of the main terms in the integral moments of various families of primitive Lfunctions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the ..."
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Cited by 13 (8 self)
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We give a newheuristic for all of the main terms in the integral moments of various families of primitive Lfunctions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are de ned by the appropriate group averages. This lends support to the idea that arithmetical Lfunctions have a spectral interpretation, and that their value distributions can be modeled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.