6,409 research outputs found
On the Value Distribution of Two Dirichlet L-functions
We look at the values of two Dirichlet -functions at the Riemann zeros (or
a horizontal shift of them). Off the critical line we show that for a positive
proportion of these points the pairs of values of the two -functions are
linearly independent over , which, in particular, means that their
arguments are different. On the critical line we show that, up to height ,
the values are different for of the Riemann zeros for some positive .Comment: 20 page
The distribution of values of the Poincare pairing for hyperbolic Riemann surfaces
For a cocompact group of SL_2(R) we fix a non-zero harmonic 1-form
\a. We normalize and order the values of the Poincare pairing
according to the length of the corresponding closed geodesic l(gamma). We prove
that these normalized values have a Gaussian distribution.Comment: 15 pages, To appear in Crelle Journa
Dissolving cusp forms: Higher order Fermi's Golden Rules
For a hyperbolic surface embedded eigenvalues of the Laplace operator are
unstable and tend to become resonances. A sufficient dissolving condition was
identified by Phillips-Sarnak and is elegantly expressed in Fermi's Golden
Rule. We prove formulas for higher approximations and obtain necessary and
sufficient conditions for dissolving a cusp form with eigenfunction into
a resonance. In the framework of perturbations in character varieties, we
relate the result to the special values of the -series . This is the Rankin-Selberg convolution of with , where
is the antiderivative of a weight 2 cusp form. In an example we show
that the above-mentioned conditions force the embedded eigenvalue to become a
resonance in a punctured neighborhood of the deformation space.Comment: 33 pages, typos corrected, new section adde
Discrete logarithms in free groups
For the free group on n generators we prove that the discrete logarithm is
distributed according to the standard Gaussian when the logarithm is
renormalized appropriately.Comment: 9 pages, 1 figure, Corrects a mistake in the Introduction and Section
Quantum Limits of Eisenstein Series and Scattering states
We identify the quantum limits of scattering states for the modular surface.
This is obtained through the study of quantum measures of non-holomorphic
Eisenstein series away from the critical line. We provide a range of stability
for the quantum unique ergodicity theorem of Luo and Sarnak.Comment: 12 pages, Corrects a typo and its ramification from previous versio
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