# parametric rmt , discrete symmetries, and cross-correlations between l -functions

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Parametric RMT , discrete symmetries, and cross-correlations between L -functions. Igor Smolyarenko Cavendish Laboratory. Collaborators: B. D. Simons, B. Conrey. July 12, 2004. - PowerPoint PPT PresentationTRANSCRIPT

Parametric RMT, discrete symmetries, and cross-correlations between L-functionsCollaborators: B. D. Simons, B. ConreyIgor Smolyarenko

Cavendish LaboratoryJuly 12, 2004

the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies. (S. Banach)Pair correlations of zeta zeros: GUE and beyond

Analogy with dynamical systems

Cross-correlations between different chaotic spectra

Cross-correlations between zeros of different (Dirichlet) L-functions

Analogy: Dynamical systems with discrete symmetries

Conclusions: conjectures and fantasies

Pair correlations of zeros Not much, really However, Montgomery 73:()As T 1 How much does the universal GUE formula tell us about the (conjectured) underlying Riemann operator?universal GUE behaviorData: M. Rubinstein

Beyond GUE: aim is nothing , but the movement is everything" Berry86-91; Keating 93; Bogomolny, Keating 96; Berry, Keating 98-99:and similarly for any Dirichlet L-function with Non-universal (lower order in) features of the pair correlation function contain a lot of information How can this information be extracted?

Poles and zerosDiscussion of the poles and zeros; the meaning of leading vs. subleading terms The pole of zeta at 1 Low-lying critical (+ trivial) zeros turn out to be connected to the classical analogue of Riemann dynamicsWhat about the rest of the structure of (1+i)?

Quantum mechanics ofclassically chaotic systems: spectral determinants and their derivatives Number theory: zeros of (1/2+i) and L(1/2+i, )Dynamiczeta-functionregularized modes of

(Perron-Frobenius spectrum)via supersymmetric nonlinear -modelStatistics of (E)Classical spectraldeterminantvia periodic orbit theoryStatistics of zerosAndreev, Altshuler, AgamBerry, Bogomolny, Keating(1+i)Periodic orbitsPrime numbersNumber theory vs. chaotic dynamicsDictionary:

Generic chaotic dynamical systems:periodic orbits and Perron-Frobenius modesZ(i) analogue of the -function on the Re s =1 line Number theory: zeros, arithmetic information, but the underlying operators are not known Chaotic dynamics: operator (Hamiltonian) is known, but not the statistics of periodic orbits Cf.:Correlation functions for chaotic spectra (under simplifying assumptions):(1-i) becomes a complementary source of information about Riemann dynamics (Bogomolny, Keating, 96)

What else can be learned? In Random Matrix Theory and in theory of dynamical systems information can be extracted from parametric correlations Simplest: H H+V(X)Spectrum of HSpectrum of H=H+V Under certain conditions on V (it has to be small either in magnitude orin rank): If spectrum of H exhibits GUE (or GOE, etc.) statistics, spectra of H and H together exhibit descendant parametric statisticsXInverse problem: given two chaotic spectra, parametric correlations can be used to extractinformation about V=H-H

Can pairs of L-functionsbe viewed as related chaotic spectra?Bogomolny, Leboeuf, 94; Rudnick and Sarnak, 98:No cross-correlations to the leading order inUsing Rubinsteins data on zeros of Dirichlet L-functions:Cross-correlation function between L(s,8) and L(s,-8):1.00.81.2R11()

Examples of parametric spectral statisticsBeyond the leading Parametric GUE terms:Analogue of the diagonal contribution(*)(*) Simons, Altshuler, 93-- norm of VPerron-FrobeniusmodesR11(x0.2)R2

Cross-correlations between L-function zeros:analytical resultsDiagonal contribution:Off-diagonal contribution:Convergent product over primesBeing computedL(1-i) is regular at 1 consistent with the absence of a leading term

Dynamical systems with discrete symmetriesSpectrum can be split into two parts, corresponding tosymmetricand antisymmetriceigenfunctionsConsider the simplest possible discrete groupIf H is invariant under G:then

Discrete symmetries: Beyond Parametric GUEConsider two irreducible representations 1 and 2 of GThe cross-correlation between the spectra of P1HP1 and P2HP2 Define P1 and P2 projection operators onto subspaces which transform according to 1 and 2are given by the analog of the dynamical zeta-function formed by projecting Perron-Frobenius operator onto subspace of the phase space which transforms according to!!

Quantum mechanics ofclassically chaotic systems: spectral determinants and their derivatives Number theory: zeros of L(1/2+i,1) and L(1/2+i, 2)DynamicL-functionregularized modes ofvia supersymmetric nonlinear -modelCorrelationsbetween 1(E) and 2(E+)Classical spectraldeterminantvia periodic orbit theoryCross-correlations of zerosL(1-i,12)Periodic orbitsPrime numbersNumber theory vs. chaotic dynamics II:Cross-correlations

The (incomplete?) to do list 0. Finish the calculation and compare to numerical data Find the correspondence betweenand the eigenvalues of information on analogues of ? Generalize to L-functions of degree > 1

Why speak about dynamics?

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