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. 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

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)?
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

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. 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”

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 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

9. 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

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