Moments of Quadratic Dirichlet L-Functions over Rational Function Fields

Abstract

We establish the meromorphic continuation of a multiple Dirichlet series associated to the fourth moment of quadratic Dirichlet L-functions, over the rational function field Fq(T) with q odd, up to its natural boundary. This is the first such result in which the group of functional equations is infinite; in such cases, it is expected that the series cannot be continued everywhere but can at least be extended to a large enough region to deduce asymptotics at the central point. In this case, these asymptotics coincide with existing predictions for the fourth moment of the symplectic family of quadratic Dirichlet Lfunctions. The construction uses the Weyl group action of a particular Kac-Moody algebra; this suggests an approach to higher moments using appropriate non-affine Kac-Moody algebras.

The second moment of quadratic twists of modular $L$-functions

“…As with Theorem 1.1 for simplicity we have restricted attention to fundamental discriminants of the form 8d, and we may adapt our methods to cover other discriminants. In families of discriminants where the sign of the functional equation is negative, we may adapt our methods to study the second moment of the derivative of the L-function at 1 2 . Further, we may adapt the technique described here to obtain an asymptotic formula for the fourth moment of quadratic Dirichlet L-functions, conditional on the GRH.…”

The fourth moment of quadratic Dirichlet L-functions over function fields

“…As q → ∞, they showed that the Frobenii classes become equidistributed in the group USp(2g), so computing the moment reduces to computing a matrix integral over USp(2g), which was done by Keating and Snaith [20]. Note that in the q → ∞ regime, Bucur and Diaconu [3] obtained an asymptotic formula for D monic deg(D)=2g L(1/2, χ D ) 4 , using multiple Dirichlet series. Hence we concentrate on the limit g → ∞ (with q fixed), which is more similar to the original number field problem.…”

Entirety of cuspidal Eisenstein series on loop groups

“…we conclude that φ(g) is bounded by the product of a −α , the sup norms δ(X)φ ∞ for any X in (A. 7), and a constant that depends only on S t . Theorem A.3.1 then follows from iterating this procedure for various simple roots α and applying the estimate in proposition A.2.5.…”

“…We note that the (presently missing) integration theory could also yield meromorphic continuations of non-constant Fourier-Whittaker coefficients. If applied to Eisenstein series on covers of loop groups, this could potentially be a source of new meromorphic continuations of multiple Dirichlet series, such as ones used for studying moments of Dirichlet L-functions (see [7], which has a detailed discussion of the fourth moment problem for quadratic characters.). be the sets of positive and negative roots, respectively.…”