197 research outputs found
Triple correlation of the Riemann zeros
We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios
of the Riemann zeta function to calculate all the lower order terms of the
triple correlation function of the Riemann zeros. A previous approach was
suggested in 1996 by Bogomolny and Keating taking inspiration from
semi-classical methods. At that point they did not write out the answer
explicitly, so we do that here, illustrating that by our method all the lower
order terms down to the constant can be calculated rigourously if one assumes
the ratios conjecture of Conrey, Farmer and Zirnbauer. Bogomolny and Keating
returned to their previous results simultaneously with this current work, and
have written out the full expression. The result presented in this paper agrees
precisely with their formula, as well as with our numerical computations, which
we include here.
We also include an alternate proof of the triple correlation of eigenvalues
from random U(N) matrices which follows a nearly identical method to that for
the Riemann zeros, but is based on the theorem for averages of ratios of
characteristic polynomials
On -gaps between zeros of the Riemann zeta-function
Under the Riemann Hypothesis, we prove for any natural number there exist
infinitely many large natural numbers such that
and
for
explicit absolute positive constants and , where
denotes an ordinate of a zero of the Riemann zeta-function on the critical
line. Selberg published announcements of this result several times but did not
include a proof. We also suggest a general framework which might lead to
stronger statements concerning the vertical distribution of nontrivial zeros of
the Riemann zeta-function.Comment: to appear in the Bulletin of the London Mathematical Societ
Bagchi's Theorem for families of automorphic forms
We prove a version of Bagchi's Theorem and of Voronin's Universality Theorem
for family of primitive cusp forms of weight and prime level, and discuss
under which conditions the argument will apply to general reasonable family of
automorphic -functions.Comment: 15 page
Open circular billiards and the Riemann hypothesis
A comparison of escape rates from one and from two holes in an experimental
container (e.g. a laser trap) can be used to obtain information about the
dynamics inside the container. If this dynamics is simple enough one can hope
to obtain exact formulas. Here we obtain exact formulas for escape from a
circular billiard with one and with two holes. The corresponding quantities are
expressed as sums over zeroes of the Riemann zeta function. Thus we demonstrate
a direct connection between recent experiments and a major unsolved problem in
mathematics, the Riemann hypothesis.Comment: 5 pages, 4 embedded postscript figures; v2: more explicit on how the
Reimann Hypothesis arises from a comparison of one and two hole escape rate
Correlations of eigenvalues and Riemann zeros
We present a new approach to obtaining the lower order terms for
-correlation of the zeros of the Riemann zeta function. Our approach is
based on the `ratios conjecture' of Conrey, Farmer, and Zirnbauer. Assuming the
ratios conjecture we prove a formula which explicitly gives all of the lower
order terms in any order correlation. Our method works equally well for random
matrix theory and gives a new expression, which is structurally the same as
that for the zeta function, for the -correlation of eigenvalues of matrices
from U(N)
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