Parametric RMT , discrete symmetries, and cross-correlations between L -functions

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

2. “…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

3. 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:

4. 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?

5. 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

6. 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)

7. 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”

8. 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

9. 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): R11() 1.2 1.0  0.8

10. 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

11. 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

12. 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

13. 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 !!

14. 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)

15. 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