A unitary test of the Ratios Conjecture

Goes, John; Jackson, Steven Glenn; Miller, Steven J.; Montague, David; Ninsuwan, Kesinee; Peckner, Ryan; Pham, Thuy Linh

doi:10.1016/j.jnt.2010.02.020

Cited by 15 publications

(19 citation statements)

References 42 publications

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“…70 Another problem is that the main term in the 1-level density agrees with random matrix theory, but the arithmetic of the family does not surface until we examine the lower order terms (which control the rate of convergence; see for example [103,107,133]). One promising line of research is the L-functions Ratios Conjecture [116,117], which is supported by corresponding calculations for random matrix ensembles (see [118,122,[125][126][127]131] for some recent work supporting these conjectures, especially [118] for a very accessible introduction to the method and a summary of its successes). Another approach is through hybrid product formulas [123].…”

Section: Future Avenuesmentioning

confidence: 98%

Nuclei, Primes and the Random Matrix Connection

Firk

Miller

2009

Symmetry

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In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mathematics -random matrix theory -provides the connection between the two fields. We assume no detailed knowledge of number theory, nuclear physics, or random matrix theory; all that is required is some familiarity with linear algebra and probability theory, as well as some results from complex analysis. Our goal is to provide the inquisitive reader with a sound overview of the subjects, placing them in their historical context in a way that is not traditionally given in the popular and technical surveys.

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Section: Future Avenuesmentioning

confidence: 98%

Nuclei, Primes and the Random Matrix Connection

Firk

Miller

2009

Symmetry

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“…Agreement has been found for suitably restricted test functions for many families. See [CS1,GJMMNPP,Mil3,Mil5,Mil6,MilMon], as well as [BCY,CS1,CS2] for agreement with other statistics. In addition to strengthening the credibility of the conjecture, these calculations provide insight into the significance of the terms that arise in the number theoretic calculations whose corresponding terms in the Ratios Conjecture's predictions are more clearly understandable.…”

Section: The Ratios Conjecturementioning

confidence: 59%

An elliptic curve test of the L-Functions Ratios Conjecture

Huynh

Miller

Morrison

2011

Journal of Number Theory

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Low lying zeros Ratios Conjecture Twists of an elliptic curve Text. We compare the L-Function Ratios Conjecture's prediction with number theory for quadratic twists of a fixed elliptic curve, showing agreement in the 1-level density up to O (X − 1−σ 2 ) for test functions supported in (−σ , σ ), giving a power-savings for σ < 1.This test introduces complications not seen in previous cases (due to the level of the elliptic curve). The results here are a key ingredients in Dueñez et al. (preprint) [DHKMS2], which determine the effective matrix size for modeling zeros near the central point.The resulting model beautifully describes the behavior of these low lying zeros for finite conductors, explaining data observed in Miller (2006) [Mil3]. A key ingredient is generalizing Jutila's bound for quadratic character sums restricted to fundamental discriminant congruent to non-zero squares modulo a square-free integer. Another application is determining the main term in the 1-level density of quadratic twists of a fixed GL n form; this generalization was implicitly assumed in Rubinstein (2001) [Rub]. Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=-Cbj1n5y-WE.

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“…The quality of this error term was tested in recent work of Fiorilli and Miller [6], who uncover new lower order terms in the one-level density for the family of Dirichlet L-functions of modulus q, and also obtain some result for the natural accuracy of the error term in the Ratios Conjecture. Other papers investigating the quality of the error term of the Ratios Conjecture include [7,11,20,21].…”

Section: Where Y (α γ ) Is Defined In (340) and A(α γ ) In (338)mentioning

confidence: 99%

One-level density of families of elliptic curves and the Ratios Conjecture

2015

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Using the Ratios Conjecture as introduced by Conrey, Farmer and Zirnbauer, we obtain closed formulas for the one-level density for two families of L-functions attached to elliptic curves, and we can then determine the underlying symmetry types of the families. The one-level scaling density for the first family corresponds to the orthogonal distribution as predicted by the conjectures of Katz and Sarnak, and the one-level scaling density for the second family is the sum of the Dirac distribution and the even orthogonal distribution. This is a new phenomenon for a family of curves with odd rank: the trivial zero at the central point accounts for the Dirac distribution, and also affects the remaining part of the scaling density which is then (maybe surprisingly) the even orthogonal distribution. The one-level density for this family was studied in the past for test functions with Fourier transforms of limited support, but since the Fourier transforms of the even orthogonal and odd orthogonal distributions are undistinguishable for small support, it was not possible to identify the distribution with those techniques. This can be done with the Ratios Conjecture, and it sheds more light on "independent" and "non-independent" zeroes, and the repulsion phenomenon.

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A unitary test of the Ratios Conjecture

Cited by 15 publications

References 42 publications

Nuclei, Primes and the Random Matrix Connection

Nuclei, Primes and the Random Matrix Connection

An elliptic curve test of the L-Functions Ratios Conjecture

One-level density of families of elliptic curves and the Ratios Conjecture

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