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