462 research outputs found
Moments of zeta and correlations of divisor-sums: III
In this series we examine the calculation of the th moment and shifted
moments of the Riemann zeta-function on the critical line using long Dirichlet
polynomials and divisor correlations. The present paper is concerned with the
precise input of the conjectural formula for the classical shifted convolution
problem for divisor sums so as to obtain all of the lower order terms in the
asymptotic formula for the mean square along of a Dirichlet polynomial
of length up to with divisor functions as coefficients
Resummation and the semiclassical theory of spectral statistics
We address the question as to why, in the semiclassical limit, classically
chaotic systems generically exhibit universal quantum spectral statistics
coincident with those of Random Matrix Theory. To do so, we use a semiclassical
resummation formalism that explicitly preserves the unitarity of the quantum
time evolution by incorporating duality relations between short and long
classical orbits. This allows us to obtain both the non-oscillatory and the
oscillatory contributions to spectral correlation functions within a unified
framework, thus overcoming a significant problem in previous approaches. In
addition, our results extend beyond the universal regime to describe the
system-specific approach to the semiclassical limit.Comment: 10 pages, no figure
Precision Tests of Parity Violation Over Cosmological Distances
Recent measurements of the Cosmic Microwave Background -mode polarization
power spectrum by the BICEP2 and POLARBEAR experiments have demonstrated new
precision tools for probing fundamental physics. Regardless of origin, the fact
that we can detect sub-K CMB polarization represents a tremendous
technological breakthrough. Yet more information may be latent in the CMB's
polarization pattern. Because of its tensorial nature, CMB polarization may
also reveal parity-violating physics via a detection of cosmic polarization
rotation. Although current CMB polarimeters are sensitive enough to measure one
degree-level polarization rotation with statistical significance,
they lack the ability to differentiate this effect from a systematic
instrumental polarization rotation. Here, we motivate the search for cosmic
polarization rotation from current CMB data as well as independent radio galaxy
and quasar polarization measurements. We argue that an improvement in
calibration accuracy would allow the precise measurement of parity- and
Lorentz-violating effects. We describe the CalSat space-based polarization
calibrator that will provide stringent control of systematic polarization angle
calibration uncertainties to -- an order of magnitude improvement
over current CMB polarization calibrators. CalSat-based calibration could be
used with current CMB polarimeters searching for -mode polarization,
effectively turning them into probes of cosmic parity violation, i.e. without
the need to build dedicated instruments.Comment: 11 pages, 3 figure
Mean Value Theorems for L-functions over Prime Polynomials for the Rational Function Field
The first and second moments are established for the family of quadratic
Dirichlet --functions over the rational function field at the central point
where the character is defined by the Legendre symbol
for polynomials over finite fields and runs over all monic irreducible
polynomials of a given odd degree. Asymptotic formulae are derived for
fixed finite fields when the degree of is large. The first moment obtained
here is the function field analogue of a result due to Jutila in the
number--field setting. The approach is based on classical analytical methods
and relies on the use of the analogue of the approximate functional equation
for these --functions.Comment: 17 page
Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts
In this note we give a combinatorial and non-computational proof of the
asymptotics of the integer moments of the moments of the characteristic
polynomials of Haar distributed unitary matrices as the size of the matrix goes
to infinity. This is achieved by relating these quantities to a lattice point
count problem. Our main result is a new explicit expression for the leading
order coefficient in the asymptotic as a volume of a certain region involving
continuous Gelfand-Tsetlin patterns with constraints.Comment: Minor improvements throughout. To appear RMT
The Classical Compact Groups and Gaussian Multiplicative Chaos
We consider powers of the absolute value of the characteristic polynomial of
Haar distributed random orthogonal or symplectic matrices, as well as powers of
the exponential of its argument, as a random measure on the unit circle minus
small neighborhoods around . We show that for small enough powers and
under suitable normalization, as the matrix size goes to infinity, these random
measures converge in distribution to a Gaussian multiplicative chaos measure.
Our result is analogous to one on unitary matrices previously established by
Christian Webb in [31]. We thus complete the connection between the classical
compact groups and Gaussian multiplicative chaos. To prove this we establish
appropriate asymptotic formulae for Toeplitz and Toeplitz+Hankel determinants
with merging singularities. Using a recent formula communicated to us by Claeys
et al., we are able to extend our result to the whole of the unit circle.Comment: 63 pages, 7 figue
Joint moments of higher order derivatives of CUE characteristic polynomials II: Structures, recursive relations, and applications
In a companion paper \cite{jon-fei}, we established asymptotic formulae for
the joint moments of derivatives of the characteristic polynomials of CUE
random matrices. The leading order coefficients of these asymptotic formulae
are expressed as partition sums of derivatives of determinants of Hankel
matrices involving I-Bessel functions, with column indices shifted by Young
diagrams. In this paper, we continue the study of these joint moments and
establish more properties for their leading order coefficients, including
structure theorems and recursive relations. We also build a connection to a
solution of the -Painlev\'{e} III equation. In the process, we give
recursive formulae for the Taylor coefficients of the Hankel determinants
formed from I-Bessel functions that appear and find differential equations that
these determinants satisfy. The approach we establish is applicable to
determinants of general Hankel matrices whose columns are shifted by Young
diagrams.Comment: 49 page
Joint moments of higher order derivatives of CUE characteristic polynomials I: asymptotic formulae
We derive explicit asymptotic formulae for the joint moments of the -th
and -th derivatives of the characteristic polynomials of CUE random
matrices for any non-negative integers . These formulae are expressed
in terms of determinants whose entries involve modified Bessel functions of the
first kind. We also express them in terms of two types of combinatorial sums.
Similar results are obtained for the analogue of Hardy's -function. We use
these formulae to formulate general conjectures for the joint moments of the
-th and -th derivatives of the Riemann zeta-function and of Hardy's
-function. Our conjectures are supported by comparison with results obtained
previously in the number theory literature.Comment: 29 page
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