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Multiple Dirichlet Series for Affine Weyl Groups
Let be the Weyl group of a simply-laced affine Kac-Moody Lie group,
excepting for even. We construct a multiple Dirichlet series
, meromorphic in a half-space, satisfying a group
of functional equations. This series is analogous to the multiple Dirichlet
series for classical Weyl groups constructed by Brubaker-Bump-Friedberg,
Chinta-Gunnells, and others. It is completely characterized by four natural
axioms concerning its coefficients, axioms which come from the geometry of
parameter spaces of hyperelliptic curves. The series constructed this way is
optimal for computing moments of character sums and L-functions, including the
fourth moment of quadratic L-functions at the central point via
and the second moment weighted by the number of divisors of the conductor via
. We also give evidence to suggest that this series appears as a
first Fourier-Whittaker coefficient in an Eisenstein series on the twofold
metaplectic cover of the relevant Kac-Moody group. The construction is limited
to the rational function field , but it also describes the
-part of the multiple Dirichlet series over an arbitrary global field
Aftermath Of The Nothing
This article consists in two parts that are complementary and autonomous at the same time.
In the first one, we develop some surprising consequences of
the introduction of a new constant called Lambda in order to represent the object ``nothing"
or ``void" into a standard set theory. On a conceptual level, it allows to see sets in a new light and to give a legitimacy to the empty set. On a technical level, it leads to a relative resolution of the anomaly of the intersection of a family free of sets.
In the second part, we show the interest of introducing an operator of potentiality into a standard set theory. Among other results, this operator allows to prove the existence of a hierarchy of empty sets and to propose a solution to the puzzle of "ubiquity" of the empty set.
Both theories are presented with equi-consistency results (model and interpretation).
Here is a declaration of intent : in each case, the starting point is a conceptual questionning; the technical tools come in a second time\\[0.4cm]
\textbf{Keywords:} nothing, void, empty set, null-class, zero-order logic with quantifiers, potential, effective, empty set, ubiquity, hierarchy, equality, equality by the bottom, identity, identification
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Logical Dreams
We discuss the past and future of set theory, axiom systems and independence
results. We deal in particular with cardinal arithmetic
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