Nuclei, Primes and the Random Matrix Connection

Firk, F.W.K.; Miller, Steven J.

doi:10.3390/sym1010064

Cited by 41 publications

(31 citation statements)

References 115 publications

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“…For a sample of the literature, see [BMSW,Bru,BM,CPRW,DFK,Gao,Go,GM,Kow1,Kow2,Mi,Mil2,RSi,RuSi,Sil3,Yo2,ZK] (especially the surveys [BMSW,Kow1,RuSi]). …”

Section: Explicit Formulamentioning

confidence: 99%

Towards an ‘average’ version of the Birch and Swinnerton-Dyer conjecture

Goes

Miller

2010

Journal of Number Theory

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Text. The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s, E). Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a family. Mestre (1986) [Mes] showed the first zero occurswhere N E is the conductor of E, though we expect the correct scale to study the zeros near the central point is the significantly smaller 1/ log N E . We significantly improve on Mestre's result by averaging over a one-parameter family of elliptic curves E over Q(T ). We assume GRH, Tate's conjecture if E is not a rational surface, and either the ABC or the Square-Free Sieve Conjecture if the discriminant has an irreducible polynomial factor of degree at least 4. We find non-trivial upper and lower bounds for the average number of normalized zeros in intervals on the order of 1/ log N E (which is the expected scale). Our results may be interpreted as providing further evidence in support of the Birch and Swinnerton-Dyer conjecture, as well as the Katz-Sarnak density conjecture from random matrix theory (as the number of zeros predicted by random matrix theory lies between our upper and lower bounds). These methods may be applied to additional families of L-functions.

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“…For a sample of the literature, see [BMSW,Bru,BM,CPRW,DFK,Gao,Go,GM,Kow1,Kow2,Mi,Mil2,RSi,RuSi,Sil3,Yo2,ZK] (especially the surveys [BMSW,Kow1,RuSi]). …”

Section: Explicit Formulamentioning

confidence: 99%

Towards an ‘average’ version of the Birch and Swinnerton-Dyer conjecture

Goes

Miller

2010

Journal of Number Theory

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“…Now that the main terms have been shown to agree, it is natural to look at the lower order terms (see [FI,HKS,Mil2,Mil4,MilPe,Yo1] for some examples). We give two applications of these terms.…”

Section: Introductionmentioning

confidence: 99%

“…There are now many examples where the main term in 1-level density calculations in number theory agrees with the Katz-Sarnak conjectures (at least for suitably restricted test functions), such as all Dirichlet characters, quadratic Dirichlet characters, L(s, ψ) with ψ a character of the ideal class group of the imaginary quadratic field Q( √ −D ) (as well as other number fields), families of elliptic curves, weight k level N cuspidal newforms, symmetric powers of GL(2) L-functions, and certain families of GL(4) and GL(6) L-functions (see [DM1,DM2,FI,Gao,Gü,HR,HuMil,ILS,KaSa2,Mil1,MilPe,OS1,OS2,RR,Ro,Rub,Yo2]). Now that the main terms have been shown to agree, it is natural to look at the lower order terms (see [FI,HKS,Mil2,Mil4,MilPe,Yo1] for some examples).…”

Section: Introductionmentioning

confidence: 99%

A unitary test of the Ratios Conjecture

Goes

Jackson

Miller

et al. 2010

Journal of Number Theory

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The Ratios Conjecture of Conrey, Farmer and Zirnbauer predicts the answers to numerous questions in number theory, ranging from n-level 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 square-root cancelation. These predictions have been verified, for suitably restricted test functions, for the 1-level density of orthogonal and symplectic families of L-functions. In this paper we verify the conjecture's predictions for the unitary family of all Dirichlet $L$-functions with prime conductor; we show square-root 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.Comment: Version 2: 24 pages, provided additional details, fixed some small mistakes and expanded the exposition in place

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“…Since its inception, Random Matrix Theory has been a powerful tool in modeling highly complicated systems, with applications in statistics [24], nuclear physics [19][20][21][22][23], and number theory [12][13][14]; see [8] for a history of the development of some of these connections. An interesting problem in Random Matrix Theory is to study subensembles of real symmetric matrices by introducing additional structure.…”

Section: Introductionmentioning

confidence: 99%

Distribution of Eigenvalues of Highly Palindromic Toeplitz Matrices

2010

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Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variables chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous work (Bryc et al., Ann. Probab. 34(1):1-38, 2006; Hammond and Miller, J. Theor. Probab. 18(3):537-566, 2005) showed that the spectral measures (the density of normalized eigenvalues) converge almost surely to a universal distribution almost that of the Gaussian, independent of p. The deficit from the Gaussian distribution is due to obstructions to solutions of Diophantine equations and can be removed (see Massey et al., J. Theor. Probab. 20(3):637-662, 2007) by making the first row palindromic.In this paper we study the case where there is more than one palindrome in the first row of real symmetric Toeplitz matrices. Using the method of moments and an analysis of the resulting Diophantine equations, we show that the spectral measures converge almost surely to a universal distribution. Assuming a conjecture on the resulting Diophantine sums (which is supported by numerics and some theoretical arguments), we prove that the limiting distribution has a fatter tail than any previously seen limiting spectral measure.

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Nuclei, Primes and the Random Matrix Connection

Cited by 41 publications

References 115 publications

Towards an ‘average’ version of the Birch and Swinnerton-Dyer conjecture

Towards an ‘average’ version of the Birch and Swinnerton-Dyer conjecture

A unitary test of the Ratios Conjecture

Distribution of Eigenvalues of Highly Palindromic Toeplitz Matrices

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