631 research outputs found
Lower bounds for moments of L-functions
The moments of central values of families of L-functions have recently
attracted much attention and, with the work of Keating and Snaith, there are
now precise conjectures for their limiting values. We develop a simple method
to establish lower bounds of the conjectured order of magnitude for several
such families of L-functions. As an example we work out the case of the family
of all Dirichlet L-functions to a prime modulus
Hecke theory and equidistribution for the quantization of linear maps of the torus
We study semi-classical limits of eigenfunctions of a quantized linear
hyperbolic automorphism of the torus ("cat map"). For some values of Planck's
constant, the spectrum of the quantized map has large degeneracies. Our first
goal in this paper is to show that these degeneracies are coupled to the
existence of quantum symmetries. There is a commutative group of unitary
operators on the state-space which commute with the quantized map and therefore
act on its eigenspaces. We call these "Hecke operators", in analogy with the
setting of the modular surface.
We call the eigenstates of both the quantized map and of all the Hecke
operators "Hecke eigenfunctions". Our second goal is to study the semiclassical
limit of the Hecke eigenfunctions. We will show that they become
equidistributed with respect to Liouville measure, that is the expectation
values of quantum observables in these eigenstates converge to the classical
phase-space average of the observable.Comment: 37 pages. New title. Spelling mistake in bibliography corrected. To
appear in Duke Math.
The distribution of spacings between quadratic residues
We study the distribution of spacings between squares modulo q, where q is
square-free and highly composite, in the limit as the number of prime factors
of q goes to infinity. We show that all correlation functions are Poissonian,
which among other things, implies that the spacings between nearest neighbors,
normalized to have unit mean, have an exponential distribution.Comment: 38 pages; introduction and section 6.2 revised, references updated.
To appear in Duke Math. Journa
The fluctuations in the number of points on a hyperelliptic curve over a finite field
The number of points on a hyperelliptic curve over a field of elements
may be expressed as where is a certain character sum. We study
fluctuations of as the curve varies over a large family of hyperelliptic
curves of genus . For fixed genus and growing , Katz and Sarnak showed
that is distributed as the trace of a random unitary
symplectic matrix. When the finite field is fixed and the genus grows, we find
that the the limiting distribution of is that of a sum of independent
trinomial random variables taking the values with probabilities
and the value 0 with probability . When both the genus
and the finite field grow, we find that has a standard Gaussian
distribution.Comment: 10 pages. Final versio
Linear statistics for zeros of Riemann's zeta function
We consider a smooth counting function of the scaled zeros of the Riemann
zeta function, around height T. We show that the first few moments tend to the
Gaussian moments, with the exact number depending on the statistic considered
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