David W. Farmer
Testing the functional equations of a high-degree Euler product.
Пнд, 11/08/2010 - 16:01 — SomNambulAuthors: David W. Farmer, Nathan C. Ryan, Ralf Schmidt
Subjects: Number Theory
AbstractWe study the L-functions associated to Siegel modular forms (equivalently,
automorphic representations of ${\rm GSp}(4,\mathbb{A}_{\mathbb{Q}})$) both
theoretically and numerically. For the L-functions of degrees 10, 14, and 16 we
perform representation theoretic calculations to cast the Langlands L-function
in classical terms. We develop a precise notion of what it means to test a
conjectured functional equation for an L-function, and we apply this to the
degree 10 adjoint L-function associated to a Siegel modular form.- 1 комментарий
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Landau-Siegel zeros and zeros of the derivative of the Riemann zeta function.
Втр, 02/09/2010 - 16:01 — SomNambulAuthors: David W. Farmer, Haseo Ki
Subjects: Number Theory
AbstractWe show that if the derivative of the Riemann zeta function has sufficiently
many zeros close to the critical line, then the zeta function has many closely
spaced zeros. This gives a condition on the zeros of the derivative of the zeta
function which implies a lower bound of the class numbers of imaginary
quadratic fields.Roots of the derivative of the Riemann zeta function and of characteristic polynomials.
Ср, 02/03/2010 - 16:01 — SomNambulAuthors: David W. Farmer, Eduardo Dueñez, Sara Froehlich, Chris Hughes, Francesco Mezzadri, Toan Phan
Subjects: Number Theory
AbstractWe investigate the horizontal distribution of zeros of the derivative of the
Riemann zeta function and compare this to the radial distribution of zeros of
the derivative of the characteristic polynomial of a random unitary matrix.
Both cases show a surprising bimodal distribution which has yet to be
explained. We show by example that the bimodality is a general phenomenon. For
the unitary matrix case we prove a conjecture of Mezzadri concerning the
leading order behavior, and we show that the same follows from the random
matrix conjectures for the zeros of the zeta function.Approximation by polynomials and Blaschke products having all zeros on a circle.
Втр, 02/02/2010 - 16:01 — SomNambulAuthors: David W. Farmer, Pamela Gorkin
Subjects: Complex Variables
AbstractWe show that a nonvanishing analytic function on a domain in the unit disc
can be approximated by (a scalar multiple of) a Blaschke product whose zeros
lie on a prescribed circle enclosing the domain. We also give a new proof of
the analogous classical result for polynomials. A connection is made to
universality results for the Riemann zeta function.
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