Quadratic Weyl group multiple Dirichlet series of Type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$

TL;DR

This paper establishes the meromorphic continuation and pole structure of quadratic affine Weyl group multiple Dirichlet series of type D4(1), deriving exact and asymptotic formulas for related moments of quadratic Dirichlet L-functions over function fields.

Contribution

It connects axiomatic and explicit constructions of the series, providing the first meromorphic continuation and residue calculations for type D4(1).

Findings

Meromorphic continuation to the complexified Tits cone.

Explicit polar divisor and residues computed.

Exact and asymptotic formulas for moments of quadratic Dirichlet L-functions.

Abstract

In this paper and its sequel \cite{DPP}, we investigate the precise relationship between the quadratic affine Weyl group multiple Dirichlet series in the sense of \cite{CG1, BD}, and those defined axiomatically by Whitehead \cite{White2} and \cite{White1}. In particular, we show that the axiomatic quadratic Weyl group multiple Dirichlet series of type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$ over rational function fields of odd characteristic admits meromorphic continuation to the interior of the corresponding complexified Tits cone. We shall also determine the polar divisor of this function, and compute the residue at each of its poles. As a consequence, we obtain an \emph{exact} formula for a weighted 4-th moment of quadratic Dirichlet $L$-functions over rational function fields; we shall also derive an asymptotic formula for this weighted moment that is expected to generalize to any global field.