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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

- 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”?

Q:

Not much, really… However,…

A:

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?

- 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

determinant

Andreev, Altshuler, Agam

via supersymmetric

nonlinear -model

Quantum mechanics of

classically chaotic systems:

spectral determinants

and their derivatives

Statistics

of (E)

regularized modes of

(Perron-Frobenius spectrum)

via periodic orbit

theory

Berry, Bogomolny, Keating

Dynamic

zeta-function

Periodic orbits

Prime numbers

Dictionary:

Statistics of zeros

Number theory:

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

(1+i)

- 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)

Cf.:

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)

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

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):

R11()

1.2

1.0

0.8

(*)

R11(x≈0.2)

R2

-- norm of V

Beyond the leading Parametric GUE terms:

Perron-Frobenius

modes

Analogue of the diagonal contribution

(*) Simons, Altshuler, ‘93

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

Consider the simplest possible discrete group

If H is invariant under G:

then

Spectrum can be split into two parts, corresponding to

symmetric

and antisymmetric

eigenfunctions

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:

Cross-correlations

Classical spectral

determinant

via supersymmetric

nonlinear -model

Quantum mechanics of

classically chaotic systems:

spectral determinants

and their derivatives

Correlations

between

1(E) and 2(E+)

regularized modes of

via periodic orbit

theory

“Dynamic

L-function”

Periodic orbits

Prime numbers

Cross-correlations of zeros

Number theory:

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

L(1-i,12)

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