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 Presentation

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  • 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?