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More than 41% of the zeros of the zeta function are on the critical line
 Acta Arith
"... The location of the zeros of the Riemann zeta function is one of the most fascinating subjects in number theory. In this paper we study the percent of zeros lying on the critical line. With the use of a new twopiece mollifier, we make a modest improvement on this important problem. ..."
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The location of the zeros of the Riemann zeta function is one of the most fascinating subjects in number theory. In this paper we study the percent of zeros lying on the critical line. With the use of a new twopiece mollifier, we make a modest improvement on this important problem.
Surpassing the Ratios Conjecture in the 1level density of Dirichlet Lfunctions
 ALGEBRA & NUMBER THEORY
, 2012
"... ..."
THE QUADRATIC CHARACTER EXPERIMENT
, 2008
"... 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 ..."
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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.
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
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
A UNITARY TEST OF THE RATIOS CONJECTURE
, 2009
"... The Ratios Conjecture of Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] predicts the answers to numerous questions in number theory, ranging from nlevel densities and correlations to mollifiers to moments and vanishing at the central point. The conjecture gives a recipe to generate these answers, which ..."
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Cited by 3 (2 self)
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The Ratios Conjecture of Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] predicts the answers to numerous questions in number theory, ranging from nlevel densities and correlations to mollifiers to moments and vanishing at the central point. The conjecture gives a recipe to generate these answers, which are believed to be correct up to squareroot cancelation. These predictions have been verified, for suitably restricted test functions, for the 1level density of orthogonal [Mil5, MilMo] and symplectic [HuyMil, Mil3, St] families of Lfunctions. In this paper we verify the conjecture’s predictions for the unitary family of all Dirichlet Lfunctions with prime conductor; we show squareroot agreement between prediction and number theory if the support of the Fourier transform of the test function is in (−1, 1), and for support up to (−2, 2) we show agreement up to a power savings in the family’s cardinality. The interesting feature in this family (which has not surfaced in previous investigations) is
An elliptic curve test of the Lfunctions Ratios Conjecture
, 2011
"... We compare the LFunction Ratios Conjecture’s prediction with number theory for the family of quadratic twists of a fixed elliptic curve with prime conductor, and show agreement in the 1level density up to an error term of size 푋 − 1−휎2 for test functions supported in (−휎, 휎); this gives us a powe ..."
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Cited by 2 (1 self)
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We compare the LFunction Ratios Conjecture’s prediction with number theory for the family of quadratic twists of a fixed elliptic curve with prime conductor, and show agreement in the 1level density up to an error term of size 푋 − 1−휎2 for test functions supported in (−휎, 휎); this gives us a powersavings for 휎 < 1. This test of the Ratios Conjecture introduces complications not seen in previous cases (due to the level of the elliptic curve). Further, the results here are one of the key ingredients in the companion paper [DHKMS2], where they are used to determine the effective matrix size for modeling zeros near the central point for this family. The resulting model beautifully describes the behavior of these low lying zeros for finite conductors, explaining the data observed by Miller in [Mil3]. A key ingredient in our analysis is a generalization of Jutila’s bound for sums of quadratic characters with the additional restriction that the fundamental discriminant be congruent to a nonzero square modulo a squarefree integer 푀. This bound is needed for two purposes. The first is to analyze the terms in the explicit formula corresponding to characters raised to an odd power. The second is to determine the main term in the 1level density of quadratic twists of a fixed form on GL푛. Such an analysis was performed by Rubinstein [Rub], who implicitly assumed that Jutila’s bound held with