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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.

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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

“…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

  • Montgomery ‘73:

universal GUE behavior



As T→ 1

Data: M. Rubinstein

How much does the universal GUE formula tell us about

the (conjectured) underlying “Riemann operator”?


Not much, really… However,…


Beyond GUE: “…aim… is nothing , but the movement is everything"

Non-universal (lower order in

) features

of the pair correlation function contain a lot of information

  • Berry’86-’91; Keating ’93; Bogomolny, Keating ’96; Berry, Keating ’98-’99:

and similarly for any Dirichlet L-function with

How can this information be extracted?

Poles and zeros

  • The pole of zeta at → 1

What about the rest of the structure of (1+i)?

  • Low-lying critical (+ trivial) zeros turn out to be connected

    to the classical analogue of “Riemann dynamics”

Discussion of the poles and zeros;

the meaning of leading vs. subleading terms

Number theory vs. chaotic dynamics

Classical spectral


Andreev, Altshuler, Agam

via supersymmetric

nonlinear -model

Quantum mechanics of

classically chaotic systems:

spectral determinants

and their derivatives


of (E)

regularized modes of

(Perron-Frobenius spectrum)

via periodic orbit


Berry, Bogomolny, Keating



Periodic orbits

Prime numbers


Statistics of zeros

Number theory:

zeros of (1/2+i)and L(1/2+i, )


Generic chaotic dynamical systems:periodic orbits and Perron-Frobenius modes

  • 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

Correlation functions for chaotic spectra (under simplifying assumptions):

(Bogomolny, Keating, ’96)


Z(i) – analogue of the -function on the Re s =1 line

(1-i) becomes a complementary source of information about “Riemann dynamics”

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 ofH´=H+V

Spectrum ofH

  • If spectrum of H exhibits GUE

    (or GOE, etc.) statistics, spectra of

    H and H´ togetherexhibit “descendant”

    parametric statistics

Under certain conditions

on V (it has to be small

either in magnitude or

in rank):

Inverse problem: given two chaotic spectra,

parametric correlations can be used to extract

information 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 in

Using Rubinstein’s data on zeros of Dirichlet L-functions:

Cross-correlation function between L(s,8)and L(s,-8):





Examples of parametric spectral statistics




-- norm of V

Beyond the leading Parametric GUE terms:



Analogue of the diagonal contribution

(*) Simons, Altshuler, ‘93

Cross-correlations between L-function zeros:analytical results

Diagonal contribution:

Off-diagonal contribution:

Convergent product

over primes

Being computed

L(1-i) is regular at 1 – consistent with the absence

of a leading term

Dynamical systems with discrete symmetries

Consider the simplest possible discrete group

If H is invariant under G:


Spectrum can be split into two parts, corresponding to


and antisymmetric


Discrete symmetries: Beyond Parametric GUE

Consider two irreducible representations 1 and 2 of G

Define P1 and P2 – projection operators onto subspaces which

transform according to 1 and 2

The cross-correlation between the spectra of P1HP1 and P2HP2

are 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


Number theory vs. chaotic dynamics II:


Classical spectral


via supersymmetric

nonlinear -model

Quantum mechanics of

classically chaotic systems:

spectral determinants

and their derivatives



1(E) and 2(E+)

regularized modes of

via periodic orbit




Periodic orbits

Prime numbers

Cross-correlations of zeros

Number theory:

zeros of L(1/2+i,1)and L(1/2+i, 2)


The (incomplete?) “to do” list

0. Finish the calculation and compare to numerical data

  • Find the correspondence between

and the eigenvalues of

information on analogues of ?

  • Generalize to L-functions of degree > 1

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