Parametric RMT, discrete symmetries, and cross-correlations between L-functions Igor Smolyarenko Cavendish Laboratory Collaborators: B. D. Simons, B. Conrey July 12, 2004

Pair correlations of zeta zeros: GUE and beyond “…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”? Q: Not much, really… However,… A:

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)? Why speak about dynamics? 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)

Z(i) – analogue of the -function on the Re s =1 line 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´=H+V Spectrum of H If spectrum of H exhibits GUE (or GOE, etc.) statistics, spectra of H and H´ together exhibit “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-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): R11() 1.2 1.0  0.8

Examples of parametric spectral statistics (*) 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

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 and antisymmetric eigenfunctions

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: 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,1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