Parametric RMT, discrete symmetries, and cross-

correlations between L-functions

Collaborators: B. D. Simons, B. Conrey

Igor Smolyarenko

Cavendish Laboratory

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

Not much, really… However,…

Pair correlations of zeros Montgomery ‘73:

( )As T → 1

How much does the universal GUE formula tell us about the (conjectured) underlying “Riemann operator”?

universal GUE behavior

Data: M. Rubinstein

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

Berry’86-’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 zeros The pole of zeta at → 1

Low-lying critical (+ trivial) zeros turn out to be connected to the classical analogue of “Riemann dynamics”

What 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-function

regularized modes of

(Perron-Frobenius spectrum)

via supersymmetric nonlinear -model

Statistics of (E)

Classical spectraldeterminant

via periodic orbit theory

Statistics of zeros

Andreev, Altshuler, Agam

Berry, Bogomolny, Keating

1+i)

Periodic orbits Prime numbers

Number theory vs. chaotic dynamics

Dictionary:

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

Z(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 H Spectrum 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 statistics

X

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):

1.0

0.8

1.2

R11()

Examples of parametric spectral statistics

Beyond the leading Parametric GUE terms:

Analogue of the diagonal contribution

(*)

(*) Simons, Altshuler, ‘93

-- norm of V

Perron-Frobeniusmodes

R11(x≈0.2)

R2

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

Spectrum can be split into two parts, corresponding to

symmetric and antisymmetric

eigenfunctions

Dynamical systems with discrete symmetries

Consider the simplest possible discrete group

If H is invariant under G:

then

Discrete symmetries: Beyond Parametric GUE

Consider two irreducible representations 1 and 2 of G

The cross-correlation between the spectra of P1HP1 and P2HP2

Define P1 and P2 – projection operators onto subspaces which transform according to 1 and 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

!!

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-function”

regularized modes of

via supersymmetric nonlinear -model

Correlationsbetween

1(E) and 2(E+)

Classical spectraldeterminant

via periodic orbit theory

Cross-correlations of zerosL1-i,12)

Periodic orbits Prime numbers

Number theory vs. chaotic dynamics II:Cross-correlations

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