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Parametric RMT, discrete
symmetries, and crosscorrelations 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:
As T → 1
universal GUE behavior
(
)
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
Andreev, Altshuler, Agam
Classical spectral
determinant
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
Dictionary:
Number theory:
zeros of (1/2+i) and L(1/2+i, )
Periodic orbits
Statistics of zeros
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)
Spectrum of H
X
Spectrum of H´=H+V
Under certain conditions
 If spectrum of H exhibits GUE
on V (it hasInverse
to be small
(or GOE,
statistics,
spectra of
problem: given
two etc.)
chaotic
spectra,
either in magnitude
or correlations
H andcan
H´ together
“descendant”
parametric
be used exhibit
to extract
in rank):
parametric
information about
V=H -H 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):
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:
Analogue of the diagonal contribution
(*) Simons, Altshuler, ‘93
Perron-Frobenius
modes
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 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
Periodic orbits
Number theory:
zeros of L(1/2+i,1) and L(1/2+i, 2)
Prime numbers
Cross-correlations of zeros
“Dynamic
L-function”
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
?