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) 1. Pair correlations of zeta zeros: GUE and beyond 2. Analogy with dynamical systems 3. Cross-correlations between

different

chaotic spectra 4. Cross-correlations between zeros of

different

(Dirichlet)

L

-functions 5. Analogy: Dynamical systems with discrete symmetries 6. 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”

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)

via

periodic orbit theory

Berry, Bogomolny, Keating

Dictionary: Periodic orbits

Statistics of zeros zeros of Number theory:



(1/2+i

)

and

L(1/2+i



,



) regularized modes of (Perron-Frobenius spectrum) Dynamic zeta-function

Prime numbers ( 1+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) 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 of

H

Spectrum of

H´=H+V

Under certain conditions  If spectrum of

H

exhibits GUE on

V

(it has to be small either in

magnitude

in

rank

): or (or GOE,

H and H

´

etc.

) statistics, spectra of

parametric

exhibit “descendant” statistics

Can pairs of

L

-functions be 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

)

:

R

11 (  ) 1.2

1.0

0.8



Examples of parametric spectral statistics

(*)

R

11 (

x ≈

0.2)

R

2  Beyond the leading P arametric GUE terms: Analogue of the diagonal contribution

-- norm

of

V

Perron-Frobenius modes (*) 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

: then Spectrum can be split into two parts, corresponding to symmetric eigenfunctions and antisymmetric

Discrete symmetries: Beyond Parametric GUE

Consider two irreducible representations  1 and  2 of

G

Define P 1 and P 2 – projection operators onto subspaces which transform according to  1 and  2 The cross-correlation between the spectra of P 1 HP 1 and P 2 HP 2 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

zeros of Number theory:

L(1/2+i



,



1 ) and

L(1/2+i



,



2 ) Cross-correlations of zeros

L

( 1-i  ,  1  2 )

The (incomplete?) “to do” list

0. Finish the calculation and compare to numerical data 1. Find the correspondence between and the eigenvalues of information on analogues of ? 2. Generalize to

L

-functions of degree > 1