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Triple correlation of the Riemann zeros
"... We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semi ..."
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We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semiclassical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously if one assumes the ratios conjecture of Conrey, Farmer and Zirnbauer. Bogomolny and Keating [4] returned to their previous results simultaneously with this current work, and have written out the full expression. The result presented in this paper agrees precisely with their formula, as well as with our numerical computations, which we include here. We also include an alternate proof of the triple correlation of eigenvalues from random U(N) matrices which follows a nearly identical method to that for the Riemann zeros, but is based on
THE QUADRATIC CHARACTER EXPERIMENT
, 802
"... Abstract. A fast new algorithm is used compute the zeros of 10 6 quadratic character Lfunctions for negative fundamental discriminants with absolute value d> 10 12. These are compared to the 1level density, including various lower order terms. These terms come from, on the one hand the Explicit Fo ..."
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Abstract. A fast new algorithm is used compute the zeros of 10 6 quadratic character Lfunctions for negative fundamental discriminants with absolute value d> 10 12. These are compared to the 1level density, including various lower order terms. These terms come from, on the one hand the Explicit Formula, and on the other the Lfunctions Ratios Conjecture. The latter give a much better fit to the data, providing numerical evidence for the conjecture. 1. Introduction. Predictions. Standard conjectures [5] predict that the low lying zeros of quadratic Dirichlet Lfunctions should be distributed according to a symplectic random matrix model. To make this more precise, we’ll introduce some notation. Let χd be a real, primitive character modulo d,
A RANDOM MATRIX MODEL FOR ELLIPTIC CURVE LFUNCTIONS OF FINITE CONDUCTOR
, 2011
"... Abstract. We propose a random matrix model for families of elliptic curve Lfunctions of finite conductor. A repulsion of the critical zeros of these Lfunctions away from the center of the critical strip was observed numerically by S. J. Miller in 2006 [50]; such behaviour deviates qualitatively fr ..."
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Abstract. We propose a random matrix model for families of elliptic curve Lfunctions of finite conductor. A repulsion of the critical zeros of these Lfunctions away from the center of the critical strip was observed numerically by S. J. Miller in 2006 [50]; such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the onelevel density of eigenvalues of orthogonal matrices after appropriate rescaling). Our purpose here is to provide a random matrix model for Miller’s surprising discovery. We consider the family of even quadratic twists of a given elliptic curve. The main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra. We call this subensemble of SO(2N) the excised orthogonal ensemble. The sievingoff of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of Lfunctions implied by the formulæ of Waldspurger and KohnenZagier. The cutoff scale
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"... 1.1. Background. Lfunctions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. The fundamental importance of these functions in mathematics is supported by the fact that two of the seven Clay Mathemati ..."
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1.1. Background. Lfunctions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. The fundamental importance of these functions in mathematics is supported by the fact that two of the seven Clay Mathematics Million Dollar Millennium Problems [20] deal with properties of these functions, namely the
NUMERICAL STUDY OF THE DERIVATIVE OF THE RIEMANN ZETA FUNCTION AT ZEROS
"... Dedicated to Professor Akio Fujii on his retirement. Abstract. The derivative of the Riemann zeta function was computed numerically on several large sets of zeros at large heights. Comparisons to known and conjectured asymptotics are presented. 1. ..."
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Dedicated to Professor Akio Fujii on his retirement. Abstract. The derivative of the Riemann zeta function was computed numerically on several large sets of zeros at large heights. Comparisons to known and conjectured asymptotics are presented. 1.
ON THE ORTHOGONAL SYMMETRY OF LFUNCTIONS OF A FAMILY OF HECKE GRÖSSENCHARACTERS
"... Abstract. The family of symmetric powers of an Lfunction associated with an elliptic curve with complex multiplication has received much attention from algebraic, automorphic and padic points of view. Here we examine this family from the perspectives of classical analytic number theory and random ..."
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Abstract. The family of symmetric powers of an Lfunction associated with an elliptic curve with complex multiplication has received much attention from algebraic, automorphic and padic points of view. Here we examine this family from the perspectives of classical analytic number theory and random matrix theory, especially focusing on evidence for the symmetry type of the family. In particular, we investigate the values at the central point and give evidence that this family can be modeled by ensembles of orthogonal matrices. We prove an asymptotic formula with power savings for the average of these Lvalues, which reproduces, by a completely different method, an asymptotic formula proven by Greenberg and Villegas–Zagier. We give an upper bound for the second moment which is conjecturally too large by just one logarithm. We also give an explicit conjecture for the second moment of this family, with power savings. Finally, we compute the one level density for this family with a test function whose Fourier transform has limited support. It is known by the work of Villegas – Zagier that the subset of these Lfunctions which have even functional equations never vanish; we show to what extent this result is reflected by our analytic results.
Riemann zeros and random matrix theory
, 2009
"... In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much re ..."
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In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much recent work concerning other varieties of Lfunctions, this article will concentrate on the zeta function as the simplest example illustrating the role of random matrix theory. 1
UPPER BOUNDS FOR THE MOMENTS OF ζ ′ (ρ)
, 806
"... Abstract. Assuming the Riemann Hypothesis, we obtain an upper bound for the 2kth moment of the derivative of the Riemann zetafunction averaged over the nontrivial zeros of ζ(s) for every positive integer k. Our bounds are nearly as sharp as the conjectured asymptotic formulae for these moments. ..."
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Abstract. Assuming the Riemann Hypothesis, we obtain an upper bound for the 2kth moment of the derivative of the Riemann zetafunction averaged over the nontrivial zeros of ζ(s) for every positive integer k. Our bounds are nearly as sharp as the conjectured asymptotic formulae for these moments.
NONVANISHING OF THE SYMMETRIC SQUARE LFUNCTION AT THE CENTRAL POINT
, 803
"... Abstract. Using the mollifier method, we show that for a positive proportion of holomorphic Hecke eigenforms of level one and weight bounded by a large enough constant, the associated symmetric square Lfunction does not vanish at the central point of its critical strip. We note that our proportion ..."
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Abstract. Using the mollifier method, we show that for a positive proportion of holomorphic Hecke eigenforms of level one and weight bounded by a large enough constant, the associated symmetric square Lfunction does not vanish at the central point of its critical strip. We note that our proportion is the same as that found by other authors for other families of Lfunctions also having symplectic symmetry type. 1.