by a series (summed over monics in $\mathbb{F}_{\! q}[x]$) of the form

\begin{equation} \label{eq: MDS-vers0}
\mathscr{W}\!(\mathbf{s}; a_{\scriptscriptstyle 2}, a_{\scriptscriptstyle 1})
\;\;\;  = \sum_{\substack{m_{\scalebox{.75}{$\scriptscriptstyle 1$}}, \ldots, m_{\scalebox{.75}{$\scriptscriptstyle 4$}},
\, d - \mathrm{monic} \\ d = d_{\scalebox{.75}{$\scriptscriptstyle 0$}}^{}
d_{\scalebox{.75}{$\scriptscriptstyle 1$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}},
\; d_{\scalebox{.75}{$\scriptscriptstyle 0$}}^{} \; \mathrm{square \; free}}}
\frac{\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}} d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}
(\widehat{m}_{\scriptscriptstyle 1}
\widehat{m}_{\scriptscriptstyle 2}
\widehat{m}_{\scriptscriptstyle 3}
\widehat{m}_{\scriptscriptstyle 4})
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 2$}}}(d_{\scriptscriptstyle 0})
A(m_{\scriptscriptstyle 1}, m_{\scriptscriptstyle 2}, m_{\scriptscriptstyle 3}, m_{\scriptscriptstyle 4}, d)}
{|m_{\scriptscriptstyle 1}|^{s_{\scalebox{.75}{$\scriptscriptstyle 1$}}}
|m_{\scriptscriptstyle 2}|^{s_{\scalebox{.75}{$\scriptscriptstyle 2$}}}
|m_{\scriptscriptstyle 3}|^{s_{\scalebox{.75}{$\scriptscriptstyle 3$}}}
|m_{\scriptscriptstyle 4}|^{s_{\scalebox{.75}{$\scriptscriptstyle 4$}}}
|d|^{s_{\scalebox{.75}{$\scriptscriptstyle 5$}}}}
\end{equation}

where $\widehat{m}_{\scriptscriptstyle i}$ ($i = 1, \ldots, 4$) is the part of
$m_{\scriptscriptstyle i}$ coprime to $d_{\scriptscriptstyle 0}.$
The coefficients
A(m_{\scalebox{1.1}{$\scriptscriptstyle 1$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 2$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 3$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 4$}}, d)
are com-pletely determined by the following two conditions:

* If $p$ is monic irreducible, then

\[
A\big(p^{k_{\scalebox{.75}{$\scriptscriptstyle 1$}}}\!,\,
p^{k_{\scalebox{.75}{$\scriptscriptstyle 2$}}}\!,\,
p^{k_{\scalebox{.75}{$\scriptscriptstyle 3$}}}\!,\,
p^{k_{\scalebox{.75}{$\scriptscriptstyle 4$}}}\!, \, p^{l} \big)
% = |p|^{k_{\scriptscriptstyle 1} + \cdots + k_{r} + l}\,
%\lambda \big(k_{\scriptscriptstyle 1}, \ldots, k_{r}, l; q^{- \deg \, p}\big)
= a\big(k_{\scalebox{1.1}{$\scriptscriptstyle 1$}},
k_{\scalebox{1.1}{$\scriptscriptstyle 2$}},
k_{\scalebox{1.1}{$\scriptscriptstyle 3$}},
k_{\scalebox{1.1}{$\scriptscriptstyle 4$}}, l;
q^{- (\deg p)\slash 2}\big).
\]

* For monic
m_{\scalebox{1.1}{$\scriptscriptstyle 1$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 2$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 3$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 4$}}, d,
we have

\[
A(m_{\scalebox{1.1}{$\scriptscriptstyle 1$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 2$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 3$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 4$}}, d)
\; = \prod_{\substack{p^{k_{\scalebox{.75}{$\scriptscriptstyle i$}}} \parallel \,
m_{\scalebox{.75}{$\scriptscriptstyle i$}}\\ p^{l} \parallel \, d}}
A\big(p^{k_{\scalebox{.75}{$\scriptscriptstyle 1$}}}\!,\,
p^{k_{\scalebox{.75}{$\scriptscriptstyle 2$}}}\!,\,
p^{k_{\scalebox{.75}{$\scriptscriptstyle 3$}}}\!,\,
p^{k_{\scalebox{.75}{$\scriptscriptstyle 4$}}}\!, \, p^{l} \big)
\]

the product being taken over monic irreducibles.

— The series defining
$\mathscr{W}\!(\mathbf{s}; a_{\scriptscriptstyle 2},
a_{\scriptscriptstyle 1})$ is absolutely convergent for
$\Re(s_{\scriptscriptstyle i}) >  1,$ $i = 1, \ldots, 5.$

The proof is similar to that of <cit.>. We first show that the series (<ref>) is
absolutely con-vergent when $\Re(s_{\scriptscriptstyle i})$ is sufficiently
large. To see this,
by condition (1) and the estimate of the coefficients
$a(\mathrm{k}; u),$ we have

\[
\big|A\big(p^{k_{\scalebox{.75}{$\scriptscriptstyle 1$}}}\!,\,
p^{k_{\scalebox{.75}{$\scriptscriptstyle 2$}}}\!,\,
p^{k_{\scalebox{.75}{$\scriptscriptstyle 3$}}}\!,\,
p^{k_{\scalebox{.75}{$\scriptscriptstyle 4$}}}\!, \,
p^{k_{\scalebox{.75}{$\scriptscriptstyle 5$}}} \big)\big|
\ll_{\epsilon, q} |p|^{\epsilon |\mathrm{k}|}
\]

for every monic irreducible $p,$ the implied constant depending upon
$q$ and $\epsilon,$ but it is independent of the degree of $p.$
Choosing $n \ge 1$ such that this constant is smaller than $q^{n},$
it follows from (2) that

\[
|A(m_{\scalebox{1.1}{$\scriptscriptstyle 1$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 2$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 3$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 4$}}, d)|
< |m_{\scalebox{1.1}{$\scriptscriptstyle 1$}}
m_{\scalebox{1.1}{$\scriptscriptstyle 2$}}
m_{\scalebox{1.1}{$\scriptscriptstyle 3$}}
m_{\scalebox{1.1}{$\scriptscriptstyle 4$}}d|^{n + \epsilon}
\]

for all monic polynomials
m_{\scalebox{1.1}{$\scriptscriptstyle 1$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 2$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 3$}},
m_{\scalebox{1.1}{$\scriptscriptstyle 4$}}, d.

\[
|\mathscr{W}\!(\mathbf{s}; a_{\scriptscriptstyle 2}, a_{\scriptscriptstyle 1})|
\;\; \le \sum_{m_{\scalebox{.75}{$\scriptscriptstyle 1$}}, \ldots,
m_{\scalebox{.75}{$\scriptscriptstyle 4$}},  \, d} \,
\frac{|A(m_{\scriptscriptstyle 1}, m_{\scriptscriptstyle 2}, m_{\scriptscriptstyle 3}, m_{\scriptscriptstyle 4}, d)|}
{|m_{\scriptscriptstyle 1}|^{\scalebox{1.1}{$\scriptscriptstyle
\Re(s_{\scalebox{.75}{$\scriptscriptstyle 1$}})$}}
|m_{\scriptscriptstyle 2}|^{\scalebox{1.1}{$\scriptscriptstyle
\Re(s_{\scalebox{.75}{$\scriptscriptstyle 2$}})$}}
|m_{\scriptscriptstyle 3}|^{\scalebox{1.1}{$\scriptscriptstyle
\Re(s_{\scalebox{.75}{$\scriptscriptstyle 3$}})$}}
|m_{\scriptscriptstyle 4}|^{\scalebox{1.1}{$\scriptscriptstyle
\Re(s_{\scalebox{.75}{$\scriptscriptstyle 4$}})$}}
\Re(s_{\scalebox{.75}{$\scriptscriptstyle 5$}})$}}}
\]

we have absolute convergence as long as
$\Re(s_{\scriptscriptstyle i}) >  n + 1 + \epsilon.$ On the other hand,
the right-hand side of the last inequality decomposes as

\[
\prod_{p} \left(1 + \sum_{\mathrm{k} \ne \mathbf{0}}
\big|a\big(\mathrm{k}; |p|^{\scalebox{1.1}{$\scriptscriptstyle - 1\slash 2$}}\big)\big|
|p|^{\scalebox{1.1}{$\scriptscriptstyle -\langle \mathrm{k}, \Re({\mathbf{s}})\rangle$}} \right)
\]

\Re({\mathbf{s}}) : = (\Re(s_{\scriptscriptstyle 1}), \ldots,
\Re(s_{\scriptscriptstyle 5})),
and $\langle \cdot, \cdot\rangle$ is the scalar product on
$\mathbb{R}^{5}.$ Taking the logarithm, our asser-tion now
follows by a simple comparison with the series

\[
\sum_{p} \sum_{\mathrm{k} \ne \mathbf{0}}
|p|^{\scalebox{1.1}{$\scriptscriptstyle - \lambda|\mathrm{k}|$}}
\]

which is easily seen to be convergent when $\lambda > 1.$

For simplicity, we shall assume from now on that $q\equiv 1 \!\pmod 4.$ Following <cit.>, we can
\begin{equation}  \label{eq: MDS-vers1}
\mathscr{W}\!(\mathbf{s};  a_{\scriptscriptstyle 2}, a_{\scriptscriptstyle 1})
\; = \sum_{d = d_{\scalebox{.75}{$\scriptscriptstyle 0$}}^{}
d_{\scalebox{.75}{$\scriptscriptstyle 1$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}}}
\frac{\prod_{i = 1}^{4} L\!\left(s_{\scriptscriptstyle i} + \tfrac{1}{2},
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}} d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}\right)
\cdot \chi_{a_{\scalebox{.8}{$\scriptscriptstyle 2$}}}\!(d_{\scriptscriptstyle 0})
P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\mathbf{s}\scalebox{1.}{$\scriptscriptstyle '$};
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}}
d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})}{|d|^{s_{\scalebox{.75}{$\scriptscriptstyle 5$}}}}
\end{equation}

P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\mathbf{s}\scalebox{1.}{$\scriptscriptstyle '$};
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}} d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
\mathbf{s}\scalebox{1.}{$\scriptscriptstyle '$} : = (s_{\scriptscriptstyle 1}, \ldots, s_{\scriptscriptstyle 4})
is the Dirichlet polynomial defined by

\begin{equation} \label{eq: polyPd}
\begin{split}
P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(
s_{\scriptscriptstyle 1}, \ldots,
s_{\scriptscriptstyle 4};
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}}
d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}) \;\; = \!
& \prod_{\substack{p^{l} \parallel \, d \\ l \, \equiv \, 1 \!\!\!\!\!\pmod 2}} \,
P_{\scalebox{.95}{$\scriptscriptstyle l$}}\!\left(
|p|^{ - s_{\scalebox{.75}{$\scriptscriptstyle 1$}}}, \ldots,
|p|^{ - s_{\scalebox{.75}{$\scriptscriptstyle 4$}}}; q^{- (\deg p)\slash 2}\right) \\
& \cdot \prod_{\substack{p \, \mid \, d_{\scalebox{.75}{$\scriptscriptstyle 1$}}
\\ p^{l} \parallel \, d \\ l \, \equiv \, 0 \!\!\!\!\pmod 2}} \,
P_{\scalebox{.95}{$\scriptscriptstyle l$}}\!\left(
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}}
d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}(p) |p|^{ - s_{\scalebox{.75}{$\scriptscriptstyle 1$}}}, \ldots,
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}}
d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}(p)|p|^{ - s_{\scalebox{.75}{$\scriptscriptstyle 4$}}};
q^{- (\deg p)\slash 2}\right).
\end{split}
\end{equation}

Here $P_{\scalebox{.95}{$\scriptscriptstyle l$}}(\underline{x}; u)$ are the polynomials in Lemma <ref>. The functions
P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\mathbf{s}\scalebox{1.}{$\scriptscriptstyle '$};
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}} d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
are symmetric in all vari-ables, and by (<ref>), they satisfy a functional equation as $s_{\scriptscriptstyle i} \to  - s_{\scriptscriptstyle i}$ for each $1\le i \le 4.$ It is clear that (<ref>)
converges absolutely for arbitrary
s_{\scriptscriptstyle 1},\ldots, s_{\scriptscriptstyle 4} \in \mathbb{C} \setminus \{1\slash 2\}
as long as $s_{\scriptscriptstyle 5}$ has sufficiently large real

We can also write

\begin{equation} \label{eq: MDS-vers2}
\mathscr{W}\!(\mathbf{s}; a_{\scriptscriptstyle 2}, a_{\scriptscriptstyle 1}) \;\; = \sum_{
m_{\scalebox{.75}{$\scriptscriptstyle 1$}}
m_{\scalebox{.75}{$\scriptscriptstyle 2$}}
m_{\scalebox{.75}{$\scriptscriptstyle 3$}}
m_{\scalebox{.75}{$\scriptscriptstyle 4$}}
\, = \, n_{\scalebox{.75}{$\scriptscriptstyle 0$}}^{}
n_{\scalebox{.75}{$\scriptscriptstyle 1$}}^{\scalebox{.8}{$\scriptscriptstyle 2$}}}
\frac{L\!\left(s_{\scalebox{1.}{$\scriptscriptstyle 5$}} + \tfrac{1}{2},
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 2$}}n_{\scalebox{.8}{$\scriptscriptstyle 0$}}}\right)
\!\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}}}\!(n_{\scriptscriptstyle 0})
Q_{\underline{m}}(s_{\scalebox{1.}{$\scriptscriptstyle 5$}};
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 2$}} n_{\scalebox{.8}{$\scriptscriptstyle 0$}}})}
{|m_{\scriptscriptstyle 1}|^{s_{\scalebox{.75}{$\scriptscriptstyle 1$}}}
|m_{\scriptscriptstyle 2}|^{s_{\scalebox{.75}{$\scriptscriptstyle 2$}}}
|m_{\scriptscriptstyle 3}|^{s_{\scalebox{.75}{$\scriptscriptstyle 3$}}}
|m_{\scriptscriptstyle 4}|^{s_{\scalebox{.75}{$\scriptscriptstyle 4$}}}}
\end{equation}

where, for
\underline{m} = (m_{\scriptscriptstyle 1}, \ldots, m_{\scriptscriptstyle 4}),
the Dirichlet polynomial
Q_{\underline{m}}(s_{\scalebox{1.}{$\scriptscriptstyle 5$}};
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 2$}} n_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
is given by

\begin{equation*} \label{eq: polyQm}
Q_{\underline{m}}(s_{\scalebox{1.}{$\scriptscriptstyle 5$}};
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 2$}}  n_{\scalebox{.8}{$\scriptscriptstyle 0$}}}) \;\;\, = \!
\prod_{\substack{p^{k_{\scalebox{.75}{$\scriptscriptstyle i$}}} \parallel \, m_{\scalebox{.75}{$\scriptscriptstyle i$}}
\\ |\underline{k}| \, \equiv \, 1 \!\!\!\!\!\pmod 2}}
Q_{\underline{k}}\!\left(|p|^{ - s_{\scalebox{.75}{$\scriptscriptstyle 5$}}};
q^{-(\deg p)\slash 2}\right) \;\;\; \cdot
\prod_{\substack{p \, \mid \, n_{\scalebox{.75}{$\scriptscriptstyle 1$}}
\\ p^{k_{\scalebox{.75}{$\scriptscriptstyle i$}}} \parallel \, m_{\scalebox{.75}{$\scriptscriptstyle i$}}
\\ |\underline{k}| \, \equiv \, 0 \!\!\!\!\pmod 2}}
Q_{\underline{k}}\!\left(\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 2$}}
n_{\scalebox{.8}{$\scriptscriptstyle 0$}}}(p)
|p|^{ - s_{\scalebox{.75}{$\scriptscriptstyle 5$}}};  q^{-(\deg p)\slash 2}\right).
\end{equation*}

Again, by (<ref>), the polynomials
Q_{\underline{m}}(s_{\scriptscriptstyle 5};
\chi_{a_{\scalebox{.8}{$\scriptscriptstyle 2$}}
n_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
satisfy a functional equation as $s_{\scriptscriptstyle 5} \to  - s_{\scriptscriptstyle 5},$ and for every
$s_{\scriptscriptstyle 5} \in \mathbb{C} \setminus \{1\slash 2\},$
the series (<ref>) converges absolutely as long as all
$s_{\scriptscriptstyle 1}, \ldots, s_{\scriptscriptstyle 4}$ have
sufficiently large real parts.
— Let $\sZ\!(\xx)$
denote the function $\sW\!(\mathbf{s}; 1, 1)$ after substituting
x_{\scalebox{1.1}{$\scriptscriptstyle i$}} = q^{ - s_{\scalebox{.75}{$\scriptscriptstyle i$}}},
$i = 1, \ldots, 5$. Then the functions $\sZ\!(\xx)$ and
$\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; \sqrt{q})$
satisfy the same functional equation (<ref>).

This follows at once from (<ref>)
and the functional equation of $L(s, \chi_{d}).$

This functional equation was stated first in [16], in terms
of the vector function

\[
\pmb{\mathscr{W}}\!(\mathbf{s}) =
\frac{1}{2}\begin{pmatrix}
\mathscr{W}\!(\mathbf{s};  1, 1) +
\mathscr{W}\!(\mathbf{s}; \theta_{\scalebox{1.1}{$\scriptscriptstyle 0$}}, 1) \\
\mathscr{W}\!(\mathbf{s};  1, 1) -
\mathscr{W}\!(\mathbf{s}; \theta_{\scalebox{1.1}{$\scriptscriptstyle 0$}}, 1)\\
\mathscr{W}\!(\mathbf{s};  1, \theta_{\scalebox{1.1}{$\scriptscriptstyle 0$}}) +
\mathscr{W}\!(\mathbf{s}; \theta_{\scalebox{1.1}{$\scriptscriptstyle 0$}},
\theta_{\scalebox{1.1}{$\scriptscriptstyle 0$}}) \\
\end{pmatrix}
\]

and predated the Chinta-Gunnells action. To make the
comparison with early work easier, and because itgeneralizes
more readily to number fields (for which the Chinta-Gunnells action
is not available), we also state the vector form
of (<ref>). The vector function obtained by substituting
x_{\scalebox{1.1}{$\scriptscriptstyle i$}} = q^{ - s_{\scalebox{.75}{$\scriptscriptstyle i$}}}
($i = 1, \ldots, 5$) in $\pmb{\mathscr{W}}\!(\mathbf{s})$
satisfies the same functional equation as the vector

\[
\mathbf{\tilde{Z}}_{\sss 1}\!(\mathbf{x}; u) : =
\begin{pmatrix}
(\tilde{Z}_{\sss W})^{+}_{\sss 1}(\xx;u) \\%^{\scriptscriptstyle (\mathrm{ev})}(\xx; u) \\
(\tilde{Z}_{\sss W})^{-}_{\sss 1}(\xx;u)\\
(\tilde{Z}_{\sss W})^{+}_{\sss 1}(\e_{\sss 5}\xx; u) \\
\end{pmatrix}=V\ov{\tZ_{\sss W}}(\xx;u), \;\; \text{where }
V= \begin{pmatrix} 1 & 0 & 1 \\
0 & 1 & 0\\
1& 0 & \!\!\!-1
\end{pmatrix}
\]

namely $\mathbf{\tilde{Z}}_{\sss 1}\!(\mathbf{x}; u) =
\mathrm{M}_{w}(\mathbf{x}; u)
\mathbf{\tilde{Z}}_{\sss 1}\!(w\mathbf{x}; u)$ for all $w\in W,$
for the $1$-cocycle $\mathrm{M}_{w}(\mathbf{x}; u)$ defined on
generators by
\mathrm{M}_{\ss_{\scalebox{0.75}{$\scriptscriptstyle i$}}}\!(\xx;u)
=-\frac{1}{x_{\scalebox{0.85}{$\scriptscriptstyle i$}}^{\scalebox{0.85}{$\scriptscriptstyle 2$}}}
V \Lam_{\ss_{\scalebox{0.75}{$\scriptscriptstyle i$}}}\!(\xx;u) V^{\sss -1}$.

Expressions similar to (<ref>), (<ref>), and the conclusions of Lemma <ref>, and Proposition <ref> all still hold if instead of
\mathscr{W}\!(\mathbf{s}; a_{\scriptscriptstyle 2}, a_{\scriptscriptstyle 1})
one takes the multiple Dirichlet series constructed in the same way using the coefficients of
$Z_{\scriptscriptstyle W}^{\scriptscriptstyle \mathrm{CG}}(\mathbf{x}; u)$. More generally, one can
start with any power series $F(z, u)$, and take the multiple Dirichlet series whose $p$-parts are
F\big(|p|^{\scalebox{1.1}{$\scriptscriptstyle - \delta(\mathbf{s})$}}, |p|^{\scalebox{1.1}{$\scriptscriptstyle - 1\slash 2$}}\big)
\tilde{Z}_{\scriptscriptstyle W}\big(|p|^{- s_{\scalebox{.85}{$\scriptscriptstyle 1$}}}, \ldots, |p|^{- s_{\scalebox{.85}{$\scriptscriptstyle 5$}}}; |p|^{\scalebox{1.1}{$\scriptscriptstyle - 1\slash 2$}}\big)
$, where
$\delta(\mathbf{s}) = s_{\scriptscriptstyle 1} + s_{\scriptscriptstyle 2} + s_{\scriptscriptstyle 3}
+ s_{\scriptscriptstyle 4} + 2s_{\scriptscriptstyle 5}
It is easy to see that this multiple Dirichlet series is, in fact,

\begin{equation}  \label{eq: WMDS-var-p-parts}
\prod_{p} F\big(|p|^{\scalebox{1.1}{$\scriptscriptstyle - \delta(\mathbf{s})$}}, |p|^{\scalebox{1.1}{$\scriptscriptstyle - 1\slash 2$}}\big)
\cdot \mathscr{W}\!(\mathbf{s}; a_{\scriptscriptstyle 2}, a_{\scriptscriptstyle 1}).
\end{equation}

Choosing $F(z, u)$ so that the product converges absolutely when $\Re(\delta(\mathbf{s})) >  6$, one sees that the multiple Dirichlet series can be expressed as in (<ref>) and (<ref>) for $\Re(s_{\scriptscriptstyle i}) >  1$, $i = 1, \ldots, 5$, and that it satisfies the required functional equations. Thus to determine a canonical normalization of $\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; u) $ (or equivalently $Z_{\scriptscriptstyle W}^{\scriptscriptstyle \mathrm{CG}}(\mathbf{x}; u)$), which eventually allows us to establish the analytic properties of the corresponding multiple Dirichlet series, some additional conditions must be imposed.

There is yet another multiple Dirichlet series

\[
Z(\mathbf{x}; q) =
\sum_{\mathrm{k} \in \mathbb{N}^{r \scalebox{.93}{$\scriptscriptstyle + 1$}}}
b(\mathrm{k}; q) \mathbf{x}^{\mathrm{k}}
\]

associated with moments of $L$-functions that is worth considering; as
before, $x_{\scalebox{1.1}{$\scriptscriptstyle i$}}$ stands for
$q^{ - s_{\scalebox{.75}{$\scriptscriptstyle i$}}},$ $i =
1, \ldots,$ $r+1,$ and the coefficients $b(\mathrm{k}; q)$ are finite sums
\sum_{\lambda}
c_{\scalebox{1.}{$\scriptscriptstyle \lambda$}}\lambda
over $q$-Weil algebraic integers of weights
\nu_{\scalebox{1.05}{$\scriptscriptstyle \lambda$}} \in \mathbb{N},
subject to the following three conditions:

* Each $q$-Weil integer $\lambda$ occurs in the sum together with
all its complex conjugates.

* The coefficients
c_{\scalebox{1.}{$\scriptscriptstyle \lambda$}}
are rational numbers, and if $\lambda, \lambda'$ are conjugates over
$\mathbb{Q},$ then
c_{\scalebox{1.}{$\scriptscriptstyle \lambda$}}
\!= c_{\scalebox{1.}{$\scriptscriptstyle \lambda \scalebox{.75}{$\scriptscriptstyle '$} $}}.

* For $|\mathrm{k}| = k_{\scalebox{1.1}{$\scriptscriptstyle 1$}}
+ \cdots + k_{r \scalebox{1.1}{$\scriptscriptstyle +1$}} > 1,$ we have
|\mathrm{k}| + 2
\le  \nu_{\scalebox{1.05}{$\scriptscriptstyle \lambda$}}
\le 2|\mathrm{k}|
for all $\lambda$ occurring in the sum.

The lower bound of $\nu_{\scalebox{1.05}{$\scriptscriptstyle
\lambda$}}$ in the last condition will be referred to as
dominance. Let $\overline{\mathbb{F}}_{\! q}$
be a fixed algebraic closure of $\mathbb{F}_{\! q}.$ We require that
the multiple Dirichlet series to be of the form (<ref>), with multiplicative coeffi-cients $B(m_{\scriptscriptstyle
1}, \ldots, m_{\scalebox{1.21}{$\scriptscriptstyle r$}}, d),$ such that the following
conditions are also satisfied:

* The sub-series

\[
\sum_{\underline{k}  \in  \mathbb{N}^{r}}
b(\underline{k}, 0; q)\underline{x}^{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}
= \prod_{i = 1}^{r} \frac{1}{1 - qx_{\scalebox{1.1}{$\scriptscriptstyle i$}}}
\]

where, as before,
\underline{x} : = (x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, x_{\scalebox{1.3}{$\scriptscriptstyle r$}})
\underline{k} : = (k_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, k_{\scalebox{1.3}{$\scriptscriptstyle r$}}).
In addition,

\[
\sum_{\underline{k}  \in  \mathbb{N}^{r}}
b(\underline{k}, 1;
\underline{k}$}} = q\;\;\;
\mathrm{and}\;\;\,
\sum_{l \, \ge \, 0} b(\underline{0}, l; q) x_{r
\scalebox{1.1}{$\scriptscriptstyle +1$}}^{l}
\!= \frac{1}{1 - qx_{r  \scalebox{1.1}{$\scriptscriptstyle
\]

In particular, $B(1, \ldots, 1) = b(0, \ldots, 0; q) = 1.$

* The coefficients $b(\mathrm{k}; q^{n})$ corresponding to
$Z(\mathbf{x}; q^{n})$ over any finite field extension $\mathbb{F}_{\! q^{n}}  \subset
\overline{\mathbb{F}}_{\! q}$ of $\mathbb{F}_{\! q}$ aregiven by

\[
b(\mathrm{k}; q^{n})
= \sum_{\lambda} c_{\scalebox{1.}{$\scriptscriptstyle
\lambda$}} \lambda^{\! n}.
\]

* For every monic irreducible $p \in \mathbb{F}_{\! q}[x]$
of degree $e \ge 1,$ the coefficients
B\big(p^{k_{\scalebox{.75}{$\scriptscriptstyle 1$}}}\!,
\ldots, p^{k_{\scalebox{.88}{$\scriptscriptstyle r$}\scalebox{.75}{$\scriptscriptstyle + 1$}}} \big)
are given by
\[
B\big(p^{k_{\scalebox{.75}{$\scriptscriptstyle 1$}}}\!,
\ldots, p^{k_{\scalebox{.88}{$\scriptscriptstyle r$}\scalebox{.75}{$\scriptscriptstyle + 1$}}} \big)
= q^{e |\mathrm{k}|}\!\sum_{\lambda} c_{\scalebox{1.}{$\scriptscriptstyle \lambda$}}\lambda^{\! - e}.
\]

In [28], the first two authors established the existence of a
unique multiple Dirichlet series satisfying all these
conditions. This axiomatic construction was generalized by Whitehead [54] to simply laced affine root systems, and by Sawin [51], using geometric methods, in a more general setting.

One should notice that it is not a priori clear that this multiple
Dirichlet series satisfies a group of func-tional equations.
To see that this is indeed the case, one can adapt the proof of
<cit.>. However, in the next section,
this will be clarified when $r = 4.$

§ COMPARISON
Let notations
be as above. Our goal in this section is to compare the functions
$\mathscr{Z}\!(\mathbf{x})$ and $\tilde{Z}_{\scriptscriptstyle
W}(\mathbf{x}; \sqrt{q}),$ anddeduce from this comparison (combined
with Theorem <ref>) the
meromorphic continuation of $\mathscr{Z}\!(\mathbf{x})$ to
themaximal possible region
$|\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}}| < 1.$

One checks that the functions $\sZ\!(\mathbf{x})$ and
$\tZW(\mathbf{x}; \sqrt{q})$ satisfy the following conditions:

* Both $\sZ\!(\mathbf{x})$ and $\tZW(\mathbf{x}; \sqrt{q})$ are
holomorphic for $|x_{\scalebox{1.1}{$\scriptscriptstyle i$}}|
< 1\slash q$ ($i = 1, \ldots, 5$).

* Both
D(\mathbf{x}; \sqrt{q})\sZ\!(\mathbf{x})
D(\mathbf{x}; \sqrt{q})\tZW(\mathbf{x}; \sqrt{q})$ are power series
that converge absolutely if either$\underline{x} \in \mathbb{C}^{\scriptscriptstyle 4}$ and
$|x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}|$ is sufficiently small, or
$x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}\in \mathbb{C}$ and
$|x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}|, \ldots,
|x_{\scalebox{1.1}{$\scriptscriptstyle 4$}}|$
are sufficiently small.

* Both $\sZ\!(\mathbf{x})$ and $\tZW(\mathbf{x}; \sqrt{q})$
are symmetric in
x_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots,
x_{\scalebox{1.1}{$\scriptscriptstyle 4$}},
and satisfy the same functional equation(<ref>).

It follows from <cit.> that

\begin{equation} \label{eq: uniqueness-thm}
\sZ\!(\mathbf{x})
= C(\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}})
\tZW(\mathbf{x}; \sqrt{q})
\end{equation}

for some function $C$ of one complex variable.
— We have the equalities

\[
\mathscr{Z}\!(\mathbf{x}) =
\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; \sqrt{q})
= Z(q^{\scalebox{.95}{$\scriptscriptstyle -1\slash 2$}}\mathbf{x}; q).
\]

We first show that $\mathscr{Z}\!(\mathbf{x})$ and
$\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; \sqrt{q})$ have the same
residue at $x_{\scalebox{1.1}{$\scriptscriptstyle 5$}} = 1\slash
\sqrt{q}$ (hence
C(\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle \delta$}})
in (<ref>)is identically $1$).

From the expression (<ref>) of
\mathscr{W}\!(\mathbf{s}; a_{\scriptscriptstyle 2}, a_{\scriptscriptstyle 1}),
we see that this function has a (simple) pole at
$s_{\scriptscriptstyle 5} = \frac{1}{2}$ only if
$a_{\scriptscriptstyle 2} = 1,$ and the part contributing to this pole is

\[
\zeta\left(s_{\scalebox{1.}{$\scriptscriptstyle 5$}} + \tfrac{1}{2}\right)
\;\;\cdot\sum_{
m_{\scalebox{.75}{$\scriptscriptstyle 1$}}
m_{\scalebox{.75}{$\scriptscriptstyle 2$}}
m_{\scalebox{.75}{$\scriptscriptstyle 3$}}
m_{\scalebox{.75}{$\scriptscriptstyle 4$}}
\, = \, \square} \;
\frac{Q_{\underline{m}}(s_{\scalebox{1.}{$\scriptscriptstyle 5$}}; 1)}
{|m_{\scriptscriptstyle 1}|^{s_{\scalebox{.75}{$\scriptscriptstyle 1$}}}
|m_{\scriptscriptstyle 2}|^{s_{\scalebox{.75}{$\scriptscriptstyle 2$}}}
|m_{\scriptscriptstyle 3}|^{s_{\scalebox{.75}{$\scriptscriptstyle 3$}}}
|m_{\scriptscriptstyle 4}|^{s_{\scalebox{.75}{$\scriptscriptstyle 4$}}}}
\,= \, \zeta\left(s_{\scalebox{1.}{$\scriptscriptstyle 5$}} + \tfrac{1}{2}\right)
\, \cdot\, \prod_{p} \bigg(\sum_{|\underline{k}| \, \equiv \, 0 \!\!\!\pmod 2}
\, \frac{Q_{\underline{k}}
\!\left(|p|^{- s_{\scalebox{.75}{$\scriptscriptstyle 5$}}};
|p|^{\scriptscriptstyle -1 \slash 2}\right)}
{|p|^{k_{\scalebox{.75}{$\scriptscriptstyle 1$}}
s_{\scalebox{.75}{$\scriptscriptstyle 1$}}
+k_{\scalebox{.75}{$\scriptscriptstyle 2$}}
s_{\scalebox{.75}{$\scriptscriptstyle 2$}}
+k_{\scalebox{.75}{$\scriptscriptstyle 3$}}
s_{\scalebox{.75}{$\scriptscriptstyle 3$}}
+k_{\scalebox{.75}{$\scriptscriptstyle 4$}}
s_{\scalebox{.75}{$\scriptscriptstyle 4$}}}}\bigg).
\]

Note that the local factor of the product in the right-hand side is

\[
\sum_{|\underline{k}| \, \equiv \, 0 \!\!\!\!\pmod 2}
Q_{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}
(x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}; u)
\underline{x}^{\scalebox{1.1}{$\scriptscriptstyle \underline{k}$}}
\]

where, for a monic irreducible $p,$ we set
x_{\scalebox{1.1}{$\scriptscriptstyle i$}} \! =  |p|^{ - s_{\scalebox{.75}{$\scriptscriptstyle i$}}}
($i=1, \ldots, 5$), and $u = |p|^{\scriptscriptstyle -1 \slash 2}.$
By (<ref>), this local
factor evaluated at
x_{\scalebox{1.1}{$\scriptscriptstyle 5$}} = u
is just $R(u\underline{x};u),$ and thus

\[
\lim_{s_{\scalebox{.75}{$\scriptscriptstyle 5$}} \to \frac{1}{2}}
\!\big(1 -  q^{\scalebox{.9}{$\scriptscriptstyle \frac{1}{2}$} - s_{\scalebox{.75}{$\scriptscriptstyle 5$}}}\big)\mathscr{W}\!(\mathbf{s};  1, a_{\scriptscriptstyle 1})
= \prod_{p} R\big(
|p|^{\scalebox{.9}{$\scriptscriptstyle - \frac{1}{2}$} - s_{\scalebox{.75}{$\scriptscriptstyle 1$}}}\!,
|p|^{\scalebox{.9}{$\scriptscriptstyle - \frac{1}{2}$} - s_{\scalebox{.75}{$\scriptscriptstyle 2$}}}\!,
|p|^{\scalebox{.9}{$\scriptscriptstyle - \frac{1}{2}$} - s_{\scalebox{.75}{$\scriptscriptstyle 3$}}}\!,
|p|^{\scalebox{.9}{$\scriptscriptstyle - \frac{1}{2}$} -
s_{\scalebox{.75}{$\scriptscriptstyle 4$}}};
|p|^{\scalebox{.9}{$\scriptscriptstyle - \frac{1}{2}$}}\big).
\]

By using the formula of $R(\underline{x};u)$ in Theorem <ref>,
the product equals
R(q^{ - s_{\scalebox{.75}{$\scriptscriptstyle 1$}}}\!,
q^{ - s_{\scalebox{.75}{$\scriptscriptstyle 2$}}}\!,
q^{ - s_{\scalebox{.75}{$\scriptscriptstyle 3$}}}\!,
q^{ - s_{\scalebox{.75}{$\scriptscriptstyle 4$}}};\sqrt{q}),
that is, $\mathscr{Z}\!(\mathbf{x})$ and
\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; \sqrt{q})
have the same residue at $x_{\scalebox{1.1}{$\scriptscriptstyle 5$}} = 1\slash
\sqrt{q}.$ This shows that

\[
\mathscr{Z}\!(\mathbf{x}) =
\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; \sqrt{q}).
\]

To prove the second equality, set $u=\sqrt{q}$. It is clear that
$\tilde{Z}_{\scriptscriptstyle  W}(u\mathbf{x};u)$ satisfies
the conditions (a) and (b) inthe
previous section. The upper bound in condition (c) follows
easily from (<ref>) and the fact that the
coeffi-cients of the power series expansion of
$J(x\slash u,u,\varepsilon)$ are polynomials in
$u^{\scalebox{1.1}{$\scriptscriptstyle -1$}}$.

To show that the coefficients of
\tilde{Z}_{\scriptscriptstyle W}(u\mathbf{x};u)
satisfy the dominance condition, we have to show that
the coef-ficients $a(\mathrm{k};u)$
in (<ref>) are divisible by
$u^{\scalebox{1.1}{$\scriptscriptstyle 2$}}$ if $|\mathrm{k}|>1$.
This follows from Corollary <ref>.

The first and third conditions in (A1) can be verified using
Lemma <ref>; the generating function of
$b(\underline{k}, 1; q)$can be computed from the term
$1\vert \ss_{\sss 5}(\xx;u)$ in $\ZW(\xx;u)$.
Conditions (A2) and (A3) can be easily verified directly.
This completes the proof.

The fact that
\mathscr{Z}\!(\mathbf{x})
coincides with
Z(q^{\scalebox{.95}{$\scriptscriptstyle -1\slash 2$}}\mathbf{x}; q)
shows that our choice of the $p$-part $\wZ_{\sss W}$ for the Weyl group multiple Dirichlet series associated with the $4$-th moment of quadratic $L$-functions is canonical; this comparison was the sole reason for introducing the axiomatic multiple Dirichlet series at the end of the previous section.
The uniqueness of $Z(\mathbf{x}; q)$ implies, for example, that the multiple Dirichlet series with all the $p$-parts equal to
Z_{\scriptscriptstyle W}^{\scriptscriptstyle \mathrm{CG}}\big(|p|^{- s_{\scalebox{.85}{$\scriptscriptstyle 1$}}}, \ldots, |p|^{- s_{\scalebox{.85}{$\scriptscriptstyle 5$}}}; |p|^{\scalebox{1.1}{$\scriptscriptstyle - 1\slash 2$}}\big)
cannot satisfy the full set of conditions that $\mathscr{Z}\!(\mathbf{x})$ satisfies. In fact, the reader can check directly using
(<ref>) in Remark <ref> that condition (A3)
(i.e., the local-to-global principle)
is not satisfied for this choice of the $p$-parts.

§ APPLICATIONS

In this final section of the paper, we begin with two straightforward
consequences of Theorem <ref>. First, by taking
coefficients of the function
\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; \sqrt{q})
with respect to the variable
$x_{\scalebox{1.1}{$\scriptscriptstyle 5$}}$, we obtain an exact
for-mula for the 4-th moments of quadratic Dirichlet
$L$-functions over rational function fields, weighted bythe
polynomials $P_{\scalebox{1.1}{$\scriptscriptstyle d$}}$ defined
in Section <ref> and Lemma <ref>. Then,
from the analytic properties of
\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; \sqrt{q})
$,we also deduce an asymptotic formula for these moments,
completely analogous to that conjectured in
[31]for arbitrary moments. For the remaining
of the section, we study in some detail the secondary terms in
the asymptotic formula. In particular, our analysis will show
that all these terms are non-zero.

— For $D \ge 1,$ we have the exact formula:

\[
\sum_{\deg d \, = D} L\!\left(\tfrac{1}{2}, \chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}\right)^{4}
\!P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
= \mathrm{Coeff}_{\xi^{\scalebox{1.}{$\scriptscriptstyle D$}}}
\tilde{Z}_{\scriptscriptstyle W} (\underline{1}, \xi; \sqrt{q})
\]

P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
= P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(0, \ldots, 0; \chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}),$
and $(\underline{1},\xi): = (1, 1, 1, 1,\xi).$

If we put
\xi=q^{ - s_{\scalebox{.75}{$\scriptscriptstyle 5$}}}
then by (<ref>) we can write

\[
\mathscr{W}\!(\mathbf{s}; 1, 1\!)
\, =\,
\sum_{D \, \ge \, 0} \left\{\sum_{\deg d \, = D}
\; \prod_{i = 1}^{4} L\!\left(s_{\scriptscriptstyle i} + \tfrac{1}{2},
\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}\right)
P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\mathbf{s}\scalebox{1.}{$\scriptscriptstyle '$};
\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}) \right\}\xi^{D}.
\]

Setting $s_{\scriptscriptstyle i} = 0,$ the asserted formula follows
at once from the equality
\mathscr{Z}\!(\underline{1}, \xi)
= \tilde{Z}_{\scriptscriptstyle W} (\underline{1}, \xi; \sqrt{q}).

However, one cannot expect to have an analogue of this result for
general function fields, or number fields.For this reason, we shall also give the asymptotic formula for the
fourth moment sums that is, indeed, ex-pected to generalize to any
global field. This asymptotic formula has also the advantage of
separating thecontributions corresponding to the
singularities of the function $\mathscr{Z}\!(\underline{1}, \xi).$

For $n \ge 1,$ let, as in <cit.>,

\[
\Phi_{n}
= \left\{\sum_{i = 1}^{5}
n_{\scriptscriptstyle i}\alpha_{\scriptscriptstyle i} \in
\Phi_{\scalebox{1.2}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{1.2}{$\scriptscriptstyle +$}}:
n_{\scriptscriptstyle 5} = n
\right\}
\]

and, for $D \ge 1,$ define $Q_{n}(D, q)$ by

\[
Q_{n}(D, q)
= \lim_{\underline{x} \, \to \, \underline{1}}\bigg(\sum_{\alpha \, \in \, \Phi_{n}}
\,\sum_{\zeta^{2n} = 1}
R_{\alpha,\,  \scalebox{1.1}{$\scriptscriptstyle \zeta$}}(\underline{x}; \sqrt{q})
\zeta^{\scalebox{1.1}{$\scriptscriptstyle D$}}
\mathbf{x}^{\scalebox{1.1}{$\scriptscriptstyle D$}
\alpha \scalebox{.9}{$\scriptscriptstyle '$}\slash n}\bigg)
\]

where $\alpha' = \alpha - n \alpha_{\scriptscriptstyle 5},$ and
R_{\alpha,\,  \scalebox{1.1}{$\scriptscriptstyle \zeta$}}(\underline{x}; \sqrt{q})
is given by (<ref>). We have the following:

— For $D, N \ge 1$ and
(N + 1)^{\scalebox{1.1}{$\scriptscriptstyle -1$}} < \Theta
< N^{\scalebox{1.1}{$\scriptscriptstyle -1$}}\!,
we have the asymptotic formula

\[
\sum_{\deg d \, = D}
L\!\left(\tfrac{1}{2}, \chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}\right)^{4}
\!P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
\, = \sum_{n \, \le \, N} Q_{n}(D, q) q^{\scalebox{.8}{$\scriptscriptstyle \frac{D}{2n}$}}
\, + \, O_{\scalebox{1.1}{$\scriptscriptstyle \Theta$}, \, q}
\left(q^{\scalebox{.8}{$\scriptscriptstyle \frac{D\Theta}{2}$}}\right).
\]

The asymptotic formula follows from Theorem
<ref> and a straightforward application of the residue
theorem to the integral

\[
\frac{1}{2 \pi i}\oint_{\partial \mathscr{A}_{\scalebox{1.}{$\scriptscriptstyle \Theta$}}}
\!\frac{\mathscr{Z}\!(\underline{1},
\xi)}{\xi^{\scalebox{1.}{$\scriptscriptstyle D$} \scalebox{1.1}{$\scriptscriptstyle + 1$}}}\, d\xi
\]

\mathscr{A}_{\scalebox{1.}{$\scriptscriptstyle \Theta$}} =
\{\xi \in \mathbb{C} : q^{\scalebox{1.1}{$\scriptscriptstyle - 2$}} \le |\xi| \le
q^{\scalebox{1.1}{$\scriptscriptstyle - \Theta\slash 2$}}\}
$ — see also <cit.>.

One might be puzzled by the discrepancy of a factor of
q^{\scalebox{1.}{$\scriptscriptstyle D\slash 2$}}
in this asymptotic formula. This is simply explained by our
\mathbf{x} \to q^{\scalebox{1.}{$\scriptscriptstyle -1\slash 2$}}\mathbf{x}
of the function
\tilde{Z}_{\scriptscriptstyle W}(\mathbf{x}; \sqrt{q})
which is causing the cor-rection polynomials
d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
to be off by a factor of $|d|^{\scalebox{1.}{$\scriptscriptstyle 1\slash 2$}}.$
For instance, when
d = d_{\scalebox{1.1}{$\scriptscriptstyle 0$}}
is square-free, then
\[
d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
=  |d|^{\scalebox{1.1}{$\scriptscriptstyle - 1\slash 2$}}
\]

instead of $1.$

Notice also that Corollary <ref> and definition (<ref>)
of the Dirichlet polynomials
P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(s_{\scriptscriptstyle 1}, \ldots,
s_{\scriptscriptstyle 4}; \chi_{a_{\scalebox{.8}{$\scriptscriptstyle 1$}}
d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
imply that
P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
is non-negative for all $d$. In particular, we have the inequality

\[
\sum_{\substack{d - \mathrm{monic \; \& \; sq. \, free} \\ \deg d \,= D}}
\chi_{d}\right)^{\scalebox{1.1}{$\scriptscriptstyle 4$}}
|d|^{\scalebox{1.1}{$\scriptscriptstyle - 1\slash 2$}}
\,\le \,
\sum_{\substack{d - \mathrm{monic}\\\deg d \, = D}}
L\!\left(\tfrac{1}{2}, \chi_{d_{\scalebox{.8}{$\scriptscriptstyle
0$}}}\right)^{\scalebox{1.1}{$\scriptscriptstyle 4$}}
\!P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}}).
\]

Moreover, <cit.> predicts an asymptotic formula
for the (traditional) moment sum in the left-hand side
of the above inequality, similar to that in Theorem <ref>.
(This similarity between the two asymp-totics should still persist
when considering the analogues of these moment sums over any global field.)
For these reasons, the presence of the correction factors
P_{\scalebox{1.1}{$\scriptscriptstyle d$}}(\chi_{d_{\scalebox{.8}{$\scriptscriptstyle 0$}}})
in our fourth moment sum is harmless for all practical purposes.

That is, if an analogue of the asymptotic formula in Theorem
<ref> is proved in thenumber field setting, it would have
the same applications as the corresponding asymptotic formula for
thetraditional fourth moment sum.

§.§ An explicit formula for $Q_{n}(D, q)$

We now give explicit formulas for the terms $Q_n(D,q)$
in Theorem <ref>, following closely [31].
In loc. cit.,conjectural formulas were given for the first two
terms $Q_{\sss 1}$ and $Q_{\sss 2}$ in the asymptotics of
the $r$-th moment ofquadratic Dirichlet $L$-functions,
summed over square-free monic polynomials. It is
interesting to note thatwe obtain the same formulas, except
for the so-called “arithmetic factor” which is simpler when summing
over all monic polynomials. Here, we are able to treat all the terms
$Q_{n}(D, q)$ since the sets $\Phi_{n}$
can be ex-plicitly described
for $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$.

First, we rewrite the double sum in the formula of $Q_n(D,u^2)$
by grouping together the terms with $\pm\zeta$. De-note by
$f^{\ec}=f^+_{\sss 1}$ (resp. $f^{\oc}=f^-_{\sss 1}$) the even
(resp. odd) part of the function $f(\xx)$ with respect to the sign
func-tion $\e_{\sss 1}$, and let $\mu_{\sss k}$ be the set
of $k$-th roots of unity in $\CC$. We have

\[
Q_{n}(D, u^{\sss 2})=\frac{1}{n}
\sum_{\zeta \in \mu_{\scalebox{.75}{$\scriptscriptstyle 2n$}}
\slash \{\pm 1\}}\zeta^{\scalebox{1.1}{$\scriptscriptstyle D$}}
\!I_{n,\, \zeta}(D,u)
\]

I_n,  ζ(D,u)
=lim_x   →  1
∑_α  ∈  Φ_n
𝐱^1.1$\scriptscriptstyle D$
α.9$\scriptscriptstyle '$n
.{R(w_1.2$\scriptscriptstyle \a$;u)
f_w_.9$\scriptscriptstyle \a$^a_.70$\scriptscriptstyle D$(;u)}
|_x_.75$\scriptscriptstyle 5$ =
ζ^1.0$\scriptscriptstyle -1$(u
𝐱^α.75$\scriptscriptstyle'$)^1.0$\scriptscriptstyle -1$n
where $f_{w}=(1+ux_{\sss 5})\| w$ was defined in Lemma <ref>, and
$a_{\sss D}\in \{\mathcal{e},\mathcal{o}\}$ denotes the parity of $D$.
Although the function $R$ does not depend on $x_{\sss 5}$, we write
$R(\xx;u)$ for $R(\ux;u)$, hence $R(w\ux;u)=R(w\xx;u)$.
Notice that $R(w\xx;u)$ is even with respect to both
$\e_{\sss 1}$ and $\e_{\sss 5}$ for all $w\in W$,
and that
I_{n,-\zeta}(D, u)=(-1)^{\scalebox{1.0}{$\scriptscriptstyle D$}}
I_{n,\, \zeta}(D,u)
so the sum over $\zeta$ above is indeed well-defined.

Next, we give integral formulas for the sum in (<ref>),
from which it will be clear that the limit exists,
and it isa polynomial in $D$ of degree 10 if $n$ is odd,
and of degree 7 if $n$ is even.

§.§.§ The case $n$ odd

Let $n=2k+1$ with $k\ge 0$.
For $I\subset S:=\{1,2,3,4\},$ possibly empty, put
$w_{\scalebox{.9}{$\scriptscriptstyle I$}}=\prod_{\sss i\in I} \ss_{\sss i},$ and
$\a_{\scalebox{.9}{$\scriptscriptstyle I$}}=\sum_{i\in I} \a_{\sss i},$
with the understanding that the empty product is the identity,
and the empty sum is 0. Also, let
t=\ss_{\sss 1}\cdots\,\ss_{\sss 4}\ss_{\sss 5}
for which $t\a_{\sss 5} = \a_{\sss 5}-\dd$. Then

\Phi_{n}=\{\a_{\sss 5}+\a_{\scalebox{.9}{$\scriptscriptstyle I$}}+k\dd : I \subset S\}
\;\;\;\, \text{and} \;\;\;\,
t^{k}w_{\scalebox{.9}{$\scriptscriptstyle I$}}(\a_{\sss 5}
+\a_{\scalebox{.9}{$\scriptscriptstyle I$}}+k \dd )=\a_{\sss 5}.

Notice that $t^2$ is a translation, with
$t^2\a_{\sss i}=\a_{\sss i}+\dd$ ($i=1,\ldots, 4$),
and $t^2\a_{\sss 5}=\a_{\sss 5}-2\dd$.
Let $n=2k+1$ with $k\ge 0,$ and let $\a=\a_{\sss 5}+k\dd$
and $w_{\scalebox{1.2}{$\scriptscriptstyle \a$}}=t^{k}$. Then we
have a decomposition

\[
R(w_{\scalebox{1.2}{$\scriptscriptstyle \a$}}\xx;u)
\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}\,=\,
\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
\mathbf{x}^{\alpha\scalebox{.75}{$\scriptscriptstyle'$}})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}} =
R_{n,\, \zeta}(\ux;u) \, \cdot \prod_{1\le i\le j\le 4}
\frac{1}{1-x_{\sss i} x_{\sss j}}
\]

R_{n,\, \zeta}(\underline{1};u)=R_{n}(u^{-2\slash n}\slash \zeta^{2})
for an explicit function $R_{n}(\varrho)$. Moreover, the function
$R_{n}(\vr)$ is given by anabsolutely convergent power
series for $|\vr|<1;$ when $n=1,$ we have

\[
R_{\sss 1}(\vr)=(\vr;\vr)_{\infty}^{\sss -11}.
\]

Due to Remark <ref> below, we omit the formula
for $R_{n}(\vr)$ when $n>1.$

The assertion follows at once from the explicit formula of
$R(\ux;u)$ in Theorem <ref>.

Recall that $f_{w}=(1+ux_{\sss 5})\|w$. For $b\in \{\mathcal{e}, \mathcal{o}\}$, let
f_{n,\, \zeta}^{b}(\underline{x};u)
=f_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}}^{b}\!(\xx;u)\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}\,=\,
\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
\mathbf{x}^{\alpha\scalebox{.75}{$\scriptscriptstyle'$}})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}}
for $\a$ and $w_{\scalebox{1.2}{$\scriptscriptstyle \a$}}$
asin the previous lemma. Since $\a$ and $w_{\a}$
are fixed, we omit them from the notation
$R_{n,\,\zeta}$ and $f_{n,\,\zeta}^{b}$.
— The limit (<ref>) is given by the

\[
I_{n,\, \zeta}(D,u)=\frac{1}{4!}\frac{1}{(2\pi i)^{^{\! 4}}}\!\oint \cdots \oint
h_{n,\, \zeta,\, D}(\underline{z};u)
\frac{\prod_{1\le i < j \le 4}(z_{\scriptscriptstyle j}^{} -
z_{\scriptscriptstyle i}^{})^{^{\! 2}}
\!(1 - z_{\scriptscriptstyle i}^{} z_{\scriptscriptstyle j}^{})}
{\prod_{i=1}^{4} (1 - z_{\scriptscriptstyle i})^{^{\! 8}}}
\frac{d z_{\scriptscriptstyle 1}}{z_{\scriptscriptstyle 1}^{\scriptscriptstyle 4}} \cdots
\frac{d z_{\scriptscriptstyle 4}}{z_{\scriptscriptstyle 4}^{\scriptscriptstyle 4}}
\]

\[
h_{n,\, \zeta, \, D}(\underline{z};u)=
\frac{R_{n,\, \zeta}(\underline{z};u)}{\prod_{\sss i=1}^{\sss 4} z_{\sss i}^{\sss D\slash (2n)}} \cdot
\begin{cases}
f_{n,\, \zeta}^{\ec}(\underline{z};u)
\prod_{i=1}^{4}\dfrac{1-u}{1-u\slash z_{\sss i}} & \text{if $D$ even} \\
f_{n,\, \zeta}^{\oc}(\underline{z};u)\prod_{i=1}^{4}
z_{\sss i}^{\sss 1\slash 2} & \text{if $D$ odd}.
\end{cases}
\]

Here each path of integration encloses the point
$z_{\sss j}=1$, but not the points $z_{\sss j} = 0, u$.

Let $\a\in\Phi_{n}$ and
w_{\scalebox{1.2}{$\scriptscriptstyle \a$}}\in W
be as in the previous lemma.
By taking $\b=\a+\a_{\scalebox{.9}{$\scriptscriptstyle I$}}\in\Phi_{n}$ in (<ref>), with
$w_{\scalebox{1.2}{$\scriptscriptstyle \b$}}
\a$}}w_{\scalebox{.9}{$\scriptscriptstyle I$}}$, we have
\[
f_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}
w_{\scalebox{.75}{$\scriptscriptstyle I$}}}(\xx)
=f_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}}\|w_{\scalebox{.9}{$\scriptscriptstyle I$}} (\xx)=
\prod_{i\, \in \, I}\frac{1-u\slash x_{\sss i}}{1-ux_{\sss i}}
f_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}}^{\ec}\!(w_{\scalebox{.9}{$\scriptscriptstyle I$}}\xx)+\frac{1}{\xx^{\a_{\scalebox{.75}{$\scriptscriptstyle I$}}}}
f_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}}^{\oc}\!(w_{\scalebox{.9}{$\scriptscriptstyle I$}}\xx).
\]
Assuming $D$ is even (the case $D$ odd being similar), we can then
write the sum in~\eqref{eq40} as
\[
\prod_{i=1}^{4} (1-u\slash x_{\sss i})x_{\sss i}^{\sss D\slash 2}
\sum_{I\subset S}
\left.h (w_{\scalebox{.9}{$\scriptscriptstyle I$}}\xx;u)
\right\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}\,=\,
\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
\mathbf{x}^{\a_{\scalebox{0.55}{$\sss I$}}+k\alpha_{\scalebox{0.55}{$\sss S$}} })^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}}
\]
h(\xx;u)= \frac{R(w_{\scalebox{.9}{$\scriptscriptstyle \a$}}\xx;u)
f_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}}^{\ec}\!(\xx;u)}{\prod_{\sss i=1}^{\sss 4}
(1-u\slash x_{\scalebox{.9}{$\scriptscriptstyle i$}}) x_{\scalebox{.9}{$\scriptscriptstyle i$}}^{\sss D\slash (2n)}}$. Now we use
the following:
\vskip5pt
\begin{fact}\label{L9.5} --- For any function $g(\xx)$ and a fixed index $1\le j\le 4$, the transformation
$x_{\sss j}\mapsto 1\slash x_{\sss j} $ takes
\[
5$}}\,=\,c\slash x_{\scalebox{.75}{$\scriptscriptstyle j$}}^{\scalebox{.95}{$\scriptscriptstyle a$}}}
\xrightarrow{x_{\scalebox{.75}{$\scriptscriptstyle j$}}\mapsto 1\slash x_{\scalebox{.75}{$\scriptscriptstyle j$}}}
g(\ss_{\sss j} \xx)\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}
\,=\,c\slash x_{\scalebox{.75}{$\scriptscriptstyle j$}}^{\scalebox{.95}{$\scriptscriptstyle b$}}}
\]
where $a, b\in\Q$ with $a+b=1$, and $c$ is any function not
depending on $x_{\sss j}$ and $x_{\sss 5}$.
\end{fact}
Taking for $g(\xx)$ the function $h(\xx;u)$ and applying
\cite[Lemma 7.1]{DT} (the case $m=0$ of Lemma~\ref{integral} below)
yields an integral representation for the sum above.
Now one can take $\ux\to \underline{1}$ inside the integral,
giving the\linebreak above expression for $I_{n,\, \zeta}(D,u)$.
\end{proof}
From the integral representation, a standard argument (e.g.,
\cite[Prop. 7.7]{DT}) leads to the following:
\vskip5pt
\begin{cor} --- When $n$ is odd, $I_{n,\, \zeta}(D,u)$ is a
polynomial in $D$ of degree 10, with leading term given by
\[
D^{10} R_{n,\, \zeta}(\underline{1};u)
f_{n,\,\zeta}^{a_{\scalebox{.70}{$\scriptscriptstyle D$}}}(\underline{1};u)
\frac{1}{4!(2n)^{^{\! 10}}}\frac{1}{(2\pi i)^{^{\! 4}}}
\!\oint_{|t_{\scalebox{.75}{$\scriptscriptstyle 4$}}|=1}
\cdots \oint_{|t_{\scalebox{.75}{$\scriptscriptstyle 1$}}|=1}
\prod_{i=1}^{4}e^{-t_{\scalebox{.75}{$\scriptscriptstyle i$}}} \cdot
\frac{\prod_{1\le i < j \le 4} (t_{\sss i}-t_{\sss j})^{^{\! 2}}
\!(t_{\sss i}+t_{\sss j} ) }{\prod_{i=1}^4 t_{\sss i}^{\sss 8}}
dt_{\sss 1}\cdots \, dt_{\sss 4}.
\]
\end{cor}
The integral (with the factor $(2\pi i)^{-4}$ included) equals
$8\slash 1575$. When $n=1$, following~\cite[Prop. 2.1]{GHRR},
we can express the leading term of $Q_{\sss 1}(D,q)$ as
\[
D^{10} R_{1}(q^{\sss -1})
\frac{1}{2^{r}} \prod_{j=0}^{r-1} \frac{(2j)!}{(r+j)!}
\;\,\quad\;\, (r=4).
\]
It agrees with the analogous main term in the conjectural
asymptotic formula of the fourth moment over the rationals
(see [27], [24], or \cite[Theorem 1.1]{GHRR}),
apart from the ``arithmetic factor''\!, which in our case
is $R_{1}(\vr) =\qpoc{\vr}{\vr}^{\sss -11}$.

\subsubsection{The case $n$ even}
Let $n=2k$ with $k\ge 1$, and let
t=\ss_{\sss 1}\cdots\, \ss_{\sss 5}
be as before, with $t\a_{\sss 5}=\a_{\sss 5}-\dd$. Then
\[
\Phi_{n}=\{\pm \a_{\sss i} +k\dd : i\in S\}\;\;\;\, \text{and} \;\;\;\,
t^{k-1}\ss_{\sss i}t(-\a_{\sss i}+k \dd)=
t^{k-1}\ss_{\sss i} t \ss_{\sss i}(\a_{\sss i}+k \dd )= \a_{\sss 5}.
\]
\vskip5pt
\begin{lem}\label{lem_even} --- Let $n=2k$ with $k\ge 1,$ and let
$\a=-\a_{\sss 4}+k\dd\in\Phi_{n}$ and
w_{\scalebox{1.2}{$\scriptscriptstyle \a$}}
=t^{k-1}\ss_{\sss 4} t$. Then we have a decom-\linebreak position
\[
\a$}}\xx;u)\vert_{x_{\scalebox{.75}{$\scriptscriptstyle
5$}}\,=\, \zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
\mathbf{x}^{\alpha\scalebox{.9}{$\scriptscriptstyle '$}
})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}} = R_{n,\, \zeta}(\ux;u)
\,\cdot \, \frac{1}{1-x_{\sss 4}^{\sss 2}}
\prod_{j=1}^{3} \frac{1}{(1-x_{\sss 4}x_{\sss j})(1-x_{\sss 4}\slash x_{\sss j})}
\]
R_{n,\, \zeta}(\underline{1};u)=
R_{n}(u^{-2\slash n}\slash \zeta^{2})
for an explicit function $R_{n}(\varrho)$. In addition, the
function $R_{n}(\vr)$ is given by an absolutely convergent power
series for $|\vr|<1$; when $n=2,$ we have
\[
R_{2}(\vr)=\qpoc{\vr}{\vr}^{\sss -8}\qpoc{\vr}{\vr^{\sss 2}}^{\sss -6}(1-1\slash \vr)^{\sss -7}.
\]
\end{lem}
\begin{proof} The assertion follows at once from the explicit formula
of $R(\ux;u)$ in Theorem~\ref{Residue-tilde-Z-avg}.
\end{proof}
b\in \{\mathcal{e}, \mathcal{o}\}
f_{n,\, \zeta}^{b}(\underline{x};u)
= f_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}}^{b}\!(\xx;u)
\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}\,=\,
\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
\mathbf{x}^{\alpha\scalebox{.75}{$\scriptscriptstyle'$}})^{\scalebox{1.0}{$\scriptscriptstyle
-1$}\slash n}}
for $\a$ and $w_{\scalebox{1.2}{$\scriptscriptstyle \a$}}$ as in the
previous lemma.
\vskip5pt
\begin{prop} \label{p9.8} --- The limit
$I_{n,\, \zeta}(D,u)$ is given by the integral
\[
\frac{1}{2^{^{3}}3!}\frac{1}{(2\pi i)^{^{\! 4}}}\!\oint \cdots \oint
h_{n,\, \zeta,\, D}(\underline{z};u)
\frac{\prod_{1\le i< j \le 4} (z_{\scriptscriptstyle i}^{}- z_{\scriptscriptstyle j}^{})^{\mathrm{e}_{\scalebox{.65}{$\scriptscriptstyle ij$}}}
(1 - z_{\scriptscriptstyle i}^{} z_{\scriptscriptstyle j}^{})
\prod_{1\le k\le l \le 3}(1 - z_{\scriptscriptstyle k}^{}
z_{\scriptscriptstyle l}^{})
\prod_{i=1}^{3} \!z_{\scriptscriptstyle i}^{}}
{\prod_{i=1}^{4} (1 - z_{\scriptscriptstyle i})^{^{\! 8}}}
\frac{d z_{\scriptscriptstyle 1}}{z_{\scriptscriptstyle 1}^{\scriptscriptstyle 4}} \cdots
\frac{d z_{\scriptscriptstyle 4}}{z_{\scriptscriptstyle 4}^{\scriptscriptstyle 4}}
\]
\mathrm{e}_{\scalebox{.85}{$\scriptscriptstyle ij$}} = 1
or $2$ according as $j = 4$ or not, and
\[
h_{n,\,\zeta,\, D}(\underline{z};u)
=\frac{R_{n,\, \zeta}(\underline{z};u)}{z_{\sss 4}^{\sss D\slash n}}
\cdot
\begin{cases}
\prod_{i=1}^{4}\dfrac{1-u}{1-u\slash z_{\sss i}} & \text{if $D$ even} \\
f_{n,\, \zeta}^{\oc}(\underline{z};u)\prod_{i=1}^{4} z_{\sss i}^{\sss 1\slash 2} & \text{if $D$ odd}.
\end{cases}
\]
The paths of integration are as in Proposition~\ref{p9.4}.
\end{prop}
\begin{proof} Let $\a\in\Phi_{n}$ and
$w_{\scalebox{1.2}{$\scriptscriptstyle \a$}}\in W$ be as in the
previous lemma, and set
$\a^{+} \!= \ss_{\sss 4}\a\in \Phi_{n}$,
w_{\scalebox{1.2}{$\scriptscriptstyle \a^{\scalebox{.6}{$\scriptscriptstyle +$}}$}}
\!=w_{\scalebox{1.2}{$\scriptscriptstyle \a$}}\ss_{\sss 4}$.
As before, assume that $D$ is even, as the case $D$ odd
is entirely similar. We have
f_{w_{\scalebox{.9}{$\scriptscriptstyle \a^{\scalebox{.6}{$\scriptscriptstyle +$}}$}}}
\! =f_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}}\|\ss_{\sss 4}
and the two terms of the sum in~\eqref{eq40}
corresponding to $\a$ and $\a^+$ can be written as
\[
\prod_{i=1}^{4}
(1-u\slash x_{\sss i})x_{\sss i}^{\sss D\slash 2}\cdot \left(
\left. h(\xx;u)\right\eval+\left.h(\ss_{\sss 4}\xx;u)\right\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}\,=\,
\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
\mathbf{x}^{\alpha^{\scalebox{.9}{$\scriptscriptstyle +$}\scalebox{.9}{$\scriptscriptstyle '$}}})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}}
\right)
\]
=\frac{R(w_{\scalebox{.9}{$\scriptscriptstyle \a$}}\xx;u)
f_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}}^{\ec}\!(\xx;u)}
{x_{\sss 4}^{\sss D\slash n}\prod_{\sss i=1}^{\sss 4} (1-u\slash
x_{\scalebox{.9}{$\scriptscriptstyle i$}})}
and $\alpha^{+ '}$ \!denotes the sum of the first four
components of $\a^{+}$.
By Fact~\ref{L9.5}, the two terms inside the parenthesis
are interchanged by the transformation
$x_{\sss 4}\mapsto 1\slash x_{\sss 4}$. Moreover,
the function $h(\xx;u)$ is clearly symmetric
in the first three variables, and
we claim that the first term in the parenthesis is also invariant
under the transformations
x_{\sss j}\mapsto 1\slash x_{\sss j},
for $j=1,2,3$. Indeed, fixing one such\linebreak $j$, by applying again
Fact~\ref{L9.5}, we have that
\[
\left.h(\xx;u)\right\eval
\xrightarrow{x_{\scalebox{.75}{$\scriptscriptstyle j$}}\mapsto
1\slash x_{\scalebox{.75}{$\scriptscriptstyle j$}}}
\left.h(\ss_{\sss j}\xx;u) \right\eval.
\]
One checks that
h(\ss_{\sss j}\xx;u)=\frac{R(w_{\scalebox{.9}{$\scriptscriptstyle \a$}}
\ss_{\scalebox{.8}{$\scriptscriptstyle j$}}\xx;u)
f_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}
\ss_{\!\scalebox{.55}{$\scriptscriptstyle j$}}}^{\ec}\!(\xx;u)}
{x_{\sss 4}^{\sss D\slash n}\prod_{\sss i=1}^{\sss 4} (1-u\slash
x_{\scalebox{.9}{$\scriptscriptstyle i$}})}
w_{\scalebox{1.2}{$\scriptscriptstyle \a$}}
\ss_{\sss j}\a
=w_{\scalebox{1.2}{$\scriptscriptstyle \a$}}\a
= \a_{\sss 5}
our claim follows now by Lemma~\ref{L6.2}.
%$(h(\ss_{\sss j}\xx;u)-h(\xx;u))\eval=0$
%$h(\ss_{\sss j}\xx;u)$ and $h(\xx;u)$
%agree when evaluated at
%${x_{\scalebox{.9}{$\scriptscriptstyle 5$}}\,=\, \zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
% \mathbf{x}^{\alpha\scalebox{.9}{$\scriptscriptstyle '$} })^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}}$,
%proving the claim.

Using these symmetries and the lemma below with $m=3$,
we can express the sum in~\eqref{eq40} as an integral. Our
assertion follows now by taking $\ux\to 1$ inside the integral.
\end{proof}
\vskip5pt
\begin{lem}\label{integral} --- Let
a_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, a_{r}
be distinct non-zero complex numbers such that $a_{\scriptscriptstyle i}a_{\scriptscriptstyle j} \ne 1$ for all
$1\le i, j \le r.$ Suppose $h$ is a function of $r$ complex variables, holomorphic on a domain containing
\Big(a_{\scalebox{1.1}{$\scriptscriptstyle 1$}}^{\delta_{\scalebox{.9}{$\scriptscriptstyle 1$}}}\!, \ldots, a_{r}^{\delta_{r}}\Big)
for all
(\delta_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, \delta_{r}) \in \{\pm 1 \}^{r}.
For $0 \le m < r,$ define $K_{m}(\mathrm{z})$ by
\[
K_{m}(\mathrm{z}) = \frac{h(\mathrm{z})}
{\prod_{k = 1}^{m}\prod_{l = m + 1}^{r} (1 - z_{\scriptscriptstyle k}^{}
z_{\scriptscriptstyle l}^{})
(1 - z_{\scriptscriptstyle k}^{\scriptscriptstyle -1}
z_{\scriptscriptstyle l}^{})
\prod_{m + 1 \, \le \, k  \, \le \, l \, \le \, r}
(1 - z_{\scriptscriptstyle k}^{}
z_{\scriptscriptstyle l}^{})}, \quad
\mathrm{z}: = (z_{\scalebox{1.1}{$\scriptscriptstyle 1$}}, \ldots, z_{r}).
\]
Then we have
\begin{equation*}
\begin{split}
&\sum_{\sigma \, \in \, \mathbb{S}_{r}} \; \sum_{\delta_{\scalebox{.85}{$\scriptscriptstyle \sigma(i)$}}  \, = \, \pm 1}
K_{m}\Big(a_{\scalebox{.95}{$\scriptscriptstyle \sigma(1)$}}^{\delta_{\scalebox{.85}{$\scriptscriptstyle \sigma(1)$}}}\!, \ldots,
a_{\scalebox{.95}{$\scriptscriptstyle \sigma(r)$}}^{\delta_{\scalebox{.85}{$\scriptscriptstyle \sigma(r)$}}} \!\Big)\\
& = \frac{(-1)^{r(r  + 1)\slash 2}}{\left(2\pi \sqrt{-1}\right)^{r}}\!
\oint \!\cdots \!\oint
\cdot \frac{\prod_{1 \, \le \, i  \, < \, j \, \le \, r}\,
(z_{\scriptscriptstyle i} - z_{\scriptscriptstyle j})^{\mathrm{e}_{\scalebox{.65}{$\scriptscriptstyle ij$}}}
(1 - z_{\scriptscriptstyle i} z_{\scriptscriptstyle j}) \, \cdot \, \prod_{1 \, \le \, k  \, \le \, l \, \le \, m}
(1 - z_{\scriptscriptstyle k} z_{\scriptscriptstyle l})
\prod_{i = 1}^{m} z_{\scriptscriptstyle i}^{\scriptscriptstyle r - m}}
{\prod_{i,\, j = 1}^{r} \big(1 - z_{\scriptscriptstyle i}a_{\scriptscriptstyle j}\big)
\big(1 - z_{\scriptscriptstyle i}a_{\scriptscriptstyle j}^{\scriptscriptstyle -1}\big)}
\frac{d \mathrm{z}}{\mathrm{z}^{\scriptscriptstyle r}}
\end{split}
\end{equation*}
where $\mathrm{e}_{\scalebox{.85}{$\scriptscriptstyle ij$}} = 1$ or $2$ according as $i\le m$ and $j\ge m+1$ or not,
$\mathrm{z}^{r} : = z_{\scalebox{.98}{$\scriptscriptstyle 1$}}^{r} \cdots\,  z_{r}^{r},$ and where
each path of integration encloses the $a_{\scriptscriptstyle j}^{ \scriptscriptstyle \pm 1},$ but not
z_{\scriptscriptstyle j} = 0.
\end{lem}
\begin{proof} The same idea as in the proof of \cite[Lemma 7.5]{DT}.
\end{proof}
The argument in \cite[Prop. 7.7]{DT} gives the following:
\vskip5pt
\begin{cor} --- When $n$ is even, $I_{n, \, \zeta}(D,u)$ is a polynomial
of degree 7 in $D$ with leading term
\[
\frac{D^{\sss 7}}{7!n^{\sss 7}}R_{n}(\zeta^{-2}u^{-2\slash n})
f_{n,\, \zeta}^{a_{\scalebox{.70}{$\scriptscriptstyle D$}}}(\underline{1};u).
\]
\end{cor}

\begin{rem}\label{r9.3}
We omitted providing formulas for the functions
$R_{n}(\vr)$ in Lemmas~\ref{lem_odd} and~\ref{lem_even},
when $n \ge 3.$\linebreak
Instead, we will show in the next subsection that the leading term of
$Q_{n}(D,u^{\sss 2})$ can be solely expressed in\linebreak
terms of $R_{\sss 1}(\vr)$ or $R_{\sss 2}(\vr)$ (according as $n$ is odd or
even), and of the matrix $B(\xx; u)$ in Theorem~\ref{Key-ingredient}.
\end{rem}

\subsubsection{The leading term of $Q_{n}(D, q)$}
The leading term of $Q_{n}(D,q)$, as a polynomial in $D$, is
\[
\frac{D^{\sss 10}}{2^{\sss 4} n^{\sss 10}} \prod_{j=0}^{3}
\frac{(2j)!}{(4+j)!} S_n(D,\sqrt q)
\;\;\; \text{or}\;\;\;
\frac{D^{\sss 7}}{7!n^{\sss 7}} S_n(D,\sqrt q)
\]
according as $n$ is odd or even. Here we set
\[
\mu_{\scalebox{.75}{$\scriptscriptstyle 2n$}}\slash\{\pm 1\}}
\zeta^{\scalebox{1.1}{$\scriptscriptstyle D$}}
R_{n}(\zeta^{-2}u^{-2\slash n})
f_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}}^{a_{\scalebox{.70}{$\scriptscriptstyle D$}}}\!
(\underline{1},\zeta^{ -1}u^{ -1\slash n};u)
\]
where $\a\in \Phi_{n}$, $w_{\a}\in W$, and the
functions $R_{n}(\varrho)$ are those occurring
in Lemmas~\ref{lem_odd} and~\ref{lem_even}. Recall that
$a_{\scalebox{.95}{$\scriptscriptstyle D$}}\in\{\mathcal{e},
\mathcal{o}\}$ stands for the parity of $D$,
and that $f^{\mathcal{e}}=f^{\eee}+f^{\oe}$, $f^{\mathcal{o}}=f^{\eo}$
for $f\in\CC(\xx,u)_{\sss 0}$.

In what follows, we show that the functions
$S_{n}(D, \sqrt q)$ are non-zero for all $n\ge 1$
and all $D$, so that all $Q_{n}(D,q)$ are present in the
asymptotic formula in Theorem~\ref{asymptotic}.
In fact, we prove the stronger result that
$S_{n}(D, \sqrt q)$ is given by a power series in $q^{-1\slash 2n}$,
whose coefficients are either all positive or all negative,
depending on the residue of $n$ modulo $4$.

We start by giving an expression for $S_{n}(D,u)$
involving only the term for $\zeta=1$ in the summation.
\!To do so, we need a property of the function $f_{w}=(1+ux_{\sss 5})\| w$.
\vskip5pt
\begin{lem} \label{L9.11} --- For all $w\in W$, the functions
$f_{w}^{\eee}$, $f_{w}^{\eo}$ and $f_{w}^{\oe}$ are even, odd and
odd, respectively, as func-\linebreak tions of~$u$.
\end{lem}
\begin{proof}
We have
\be\label{eq41}
\ov{f_{w}}(\xx; u)=\tL_{w}(\xx; u)
\cdot \, \!^t(1, u\xx^{w^{\scalebox{.75}{$\scriptscriptstyle -1$}}
\a_{\scalebox{.75}{$\scriptscriptstyle 5$}}}\!,0).
\ee
As functions of $u$, the entries of $\tL_{w}(\xx;u)$ have the same parities
as the corresponding entries of the matrix
\left(\begin{smallmatrix} 1& u& u\\
u& 1& 1\\
u& 1& 1
\end{smallmatrix}\right)\!.
This follows easily by induction on the length $\ell(w)$ of $w$; the
case $\ell(w)=1$ is clear from the formulas of $\Lam_{\sss 1}$ and
$\Lam_{\sss 2}$ in Section~\ref{s4.1}, and the product of such matrices
is again of the same type, so the co-\linebreak cycle relation gives the induction
step. Our assertions now follow from~\eqref{eq41}.
\end{proof}
For $c\in\Z$, let $\UU_{n,\, c}$ be the operator acting on {L}aurent
series $f(\vr)=\sum_{k \, \in\, \Z} n_{\sss k} \vr^{k} $ with $n_{\sss k} \in \CC$,
%in the variable $\vr=u^{\sss -2\slash n}$
defined by\linebreak
\[
\UU_{n,\,c}(f)\;
:=\; \sum_{\substack{k \, \equiv \, c\!\!\!\!\pmod{n}}} n_{\sss k} \vr^{k}
=\;\frac{1}{n}\sum_{\zeta\in \mu_{\scalebox{.75}{$\scriptscriptstyle n$}}}\zeta^{c} f(\zeta^{\sss -1} \vr).
\]
We will apply this operator to Laurent series $f$ which
converge absolutely for $0<|\vr|<1$, when it is well-defined and the equality above clearly holds.
Note that $\UU_{n,\,c}$ depends only on~$c$ modulo $n$.
%We omit the
%dependence on~$n$ from the notation since $n$ is fixed throughout.
\vskip5pt
\begin{prop}\label{p9.12} ---
With $\a$, $w_{\a}$ and $R_{n}(\vr)$ from Lemma~\ref{lem_odd}
(resp. Lemma~\ref{lem_even}) for $n$ odd (resp. $n$ even),
we\linebreak have:
\[
S_{n}(D, u) = \UU_{n,\,{\sss D\slash 2}}\!\left[R_{n}(u^{ -2\slash n}) \cdot
f^{\eee}_{w_{\a}}(\u1, u^{ -1\slash n} ;u)\right]+
u\UU_{n,\, {\sss D\slash 2}}\!\left[R_{n}(u^{ -2\slash n} ) \cdot
u^{\sss -1}f^{\oe}_{w_{\a}}(\u1, u^{ -1\slash n} ;u) \right]
\]
if $D$ is even, and
\[
S_{n}(D, u) = u^{ 1-1\slash n}\,
\UU_{n,\, {\sss \lfloor D \slash 2\rfloor}}\!\left[R_{n }(u^{ -2\slash n})\cdot
u^{ -1+1\slash n}f^{\eo}_{w_{\a}}
(\u1,u^{ -1\slash n} ;u) \right]
\]
if $D$ is odd. Here we view the functions between brackets
as Laurent series in $\vr=u^{ -2\slash n}$, by the previous lemma.
\end{prop}
\begin{proof} By Lemma~\ref{L9.11}, we have that
\[
f^{\eee}_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}}(\underline{1}, x_{\sss 5}; u)
= g_{\sss 1}(x_{\sss 5}^{\sss 2} , u^{\sss 2}),\;\;\;\,
f^{\eo}_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}}(\underline{1}, x_{\sss 5}; u)
= ux_{\sss 5} g_{\sss 2}(x_{\sss 5}^{\sss 2} , u^{\sss 2}) \;\;\;\, \text{and}\;\;\;\,
f^{\oe}_{w_{\scalebox{.9}{$\scriptscriptstyle \a$}}}(\underline{1}, x_{\sss 5}; u) =
u g_{\sss 3}(x_{\sss 5}^{\sss 2} , u^{\sss 2})
\]
with $g_{\sss i}\in \Q(x,u)$. Thus, assuming $D$ to be even
(the case $D$ odd being similar), we have the following
ex-\linebreak pression for $S_{n}(D,u)$:
\[
\begin{aligned}
S_{n}(D, u)= S_{n}(D, \vr^{-n\slash 2})&
= \frac{1}{n}\sum_{\zeta\in\mu_{\scalebox{.75}{$\scriptscriptstyle 2n$}}
\slash \{\pm 1\}}\zeta^{\scalebox{1.1}{$\sss D$}}
R_{n}(\vr\slash \zeta^{2})\big(g_{\sss 1}(\vr\slash \zeta^{2}, \vr^{-n})
+ ug_{\sss 3}(\vr\slash \zeta^{2}, \vr^{-n})\big)\\
& = \UU_{n,\, {\sss D\slash 2}}[R_{n}(\vr) g_{\sss 1}(\vr, \vr^{-n})]
+ u\UU_{n,\, {\sss D\slash 2}} [R_{n}(\vr) g_{\sss 3}(\vr, \vr^{-n})].
\end{aligned}
\]
Note that the functions between brackets converge absolutely as
Laurent series in $\vr$ for $0<|\vr|<1$,
by Lemmas~\ref{lem_odd} and~\ref{lem_even}, so the last
equality holds. Switching back to the variable $u$
completes the proof.
\end{proof}
This proposition yields explicit formulas for the leading term
of $Q_{n}(D,q)$, which, for small $n$, can be used\linebreak
to check that this term is given by a non-zero power series
with coefficients of the same sign. To prove this\linebreak
property for all~$n$, in Corollary~\ref{C9.14} below, we shall
give different expressions for the functions between brackets
in the proposition, using the extra functional equation
introduced in Section~\ref{s4.2}.

Recall the extension of the cocycle $\tL$ to
$W\oplus \Z$ from the beginning of Section~\ref{s6.3}.
In particular, we have\linebreak
\[
\tL_{\tau^{-1}}(\xx;u)=\tB(\xx;u) = B(\xx;u)\slash
(1-u^{\sss 2}x^\dd)^{\scalebox{1.1}{$\sss 2$}}
\]
and the vector function
\mathbf{\tilde{Z}}=\ov{\tilde{Z}_{\sss W}}
satisfies the functional equation~\eqref{200}, with the cocycle
$\tL$ in place of~$\Lam$.\linebreak

\vskip-14.5pt
Consider the vector version of the residue in Lemma~\ref{L6.2}:
\[
\vC_{\alpha,\, \scalebox{1.1}{$\scriptscriptstyle \zeta$}}(\underline{x};u):=
\hskip-44pt \lim_{\hskip44pt x_{\scalebox{.75}{$\scriptscriptstyle 5$}}\to
\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
\mathbf{x}^{\alpha \scalebox{.75}{$\scriptscriptstyle '$}})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}}
\!\left(1 -\zeta u^{1\slash n}
\mathbf{x}^{\alpha \slash n}\right)
\tZZ(\xx;u)
\]
for any
\a=n\a_{\sss 5}+\a\scalebox{1.1}{$\scriptscriptstyle '$}\in\Phi_{n}
$\zeta\in\mu_{\scalebox{1.1}{$\scriptscriptstyle 2n$}}$.
The residue vector
\vC_{\alpha,\,  \scalebox{1.1}{$\scriptscriptstyle \zeta$}}(\underline{x};u)
appears in the proof of Lemma~\ref{L6.2}, where it is shown that
\be\label{49}
\vC_{\alpha,\, \scalebox{1.1}{$\scriptscriptstyle \zeta$}}(\underline{x};u)=
\frac{1}{2n} \left. \!\left\{ R(w_\a \ux; u) \ov{f_{w_\a}}(\xx;u)
\right\}\right\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}
\,=\,\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}} (u \mathbf{x}^{\alpha\scalebox{.9}{$\scriptscriptstyle '$}
})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}}
\ee
with $w_{\a}\in W$ such that $w_{\a} \a =\a_{\sss 5}$.

Assuming now that~$\a$ is chosen as in
Lemmas~\ref{lem_odd} and~\ref{lem_even}, we give a formula for
\vC_{\alpha,\,  \scalebox{1.1}{$\scriptscriptstyle
\zeta$}}(\underline{x};u)
in terms of the above extended cocycle.
\vskip5pt
\begin{lem}\label{L9.13} --- a). If $n=2k+1$ and
$\a=\a_{\sss 5}+k\dd\in\Phi_n$, then
\[
\vC_{\alpha,\, \scalebox{1.1}{$\scriptscriptstyle \zeta$}}(\underline{x};u)
= \frac{1}{2n}
\left. \!\left\{ R(\ux; u\xx^{k\dd})
\tL_{\tau^{-k}}(\xx;u) \cdot v_{\sss 0}\right\}
\right\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}
\,=\,\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}} (u \mathbf{x}^{\alpha\scalebox{.9}{$\scriptscriptstyle '$}
})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}}
\]
where $v_{\sss 0}=\, \!^t(1,1,0)$.

b). If $n=2k+2$ and $\a=-\a_{\sss 4}+(k+1)\dd \in\Phi_{n}$, then
\[
\vC_{\alpha,\, \scalebox{1.1}{$\scriptscriptstyle \zeta$}}(\underline{x};u)
= \frac{1}{2n}
\left.\!\left\{R(\ss_{\sss 4}t\ux; u\xx^{k\dd})
\tL_{\tau^{-k}}(\xx;u)
\cdot \ov{f_{\ss_{\scalebox{.67}{$\scriptscriptstyle 4$}} t}}(\xx; u\xx^{k\dd}) \right\}
\right\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}
\,=\, \zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}
(u \mathbf{x}^{\alpha\scalebox{.9}{$\scriptscriptstyle '$}
})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}}
\]
where $t=\ss_{\sss 1}\ss_{\sss 2}\ss_{\sss 3}\ss_{\sss 4} \ss_{\sss 5}$.
\end{lem}
\begin{proof}
We use the functional equation~\eqref{200}, written for
$\tZZ$ and the cocycle $\tL$, to compute the
residue vector
\vC_{\alpha,\, \scalebox{1.1}{$\scriptscriptstyle
\zeta$}}(\underline{x};u)
In part a), we take the element $\tau^{-k}$ in~\eqref{200},
while in part b) we take the element $\ss_{\sss 4}t\tau^{-k}$.
Note that by Theorem~\ref{Key-ingredient} and the cocycle relation,
the matrix $\tL_{\tau^{-k}}(\xx;u)$ has poles only if
$u\xx^{\b}=\pm 1$ for some
\b\in\Phi_{\scalebox{.95}{$\scriptscriptstyle \mathrm{re}$}}^{\scalebox{.95}{$\scriptscriptstyle +$}}
with $\b<k\dd$. Since $\a>k\dd$, it can be evaluated at
x_{\sss 5}\,=\, \zeta^{\sss -1}(u \mathbf{x}^{\alpha\scalebox{.9}{$\scriptscriptstyle '$}})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}
that is, when $u\xx^{\a}=\pm 1$.
\end{proof}
The decompositions in Lemmas~\ref{lem_odd} and~\ref{lem_even}
have the following analogues:
\vskip5pt
\begin{lem}\label{L9.14} --- a). For $n=2k+1\ge 1$
and $\a=\a_{\sss 5}+k\dd\in\Phi_{n}$, we have
\[
R(\ux; u\xx^{k\dd})
\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}
\,=\, \zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}}(u
\mathbf{x}^{\alpha\scalebox{.9}{$\scriptscriptstyle
'$}})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}}
= R_{\sss 1,\,1}\!\left(\ux;u\xx^{k\dd}\eval\right)
\, \cdot \, \prod_{1\le i\le j\le 4} \frac{1}{1-x_{\sss i} x_{\sss j}}
\]
where the first factor in the right-hand side becomes
$R_{\sss 1}(u^{-2\slash n}\slash \zeta^{2})$ when evaluated
at $\ux=\u1$. Here $R_{\sss 1,\,1}(\ux;u)$ and
$R_{\sss 1}(\vr)$ are the same functions as in Lemma~\ref{lem_odd}.

b). For $n=2k+2\ge 2$ and $\a=-\a_{\sss 4}+(k+1)\dd\in\Phi_{n}$,
we have
\[
R(\ss_{\sss 4}t\ux; u\xx^{k\dd})
\vert_{x_{\scalebox{.75}{$\scriptscriptstyle 5$}}
\,=\,\zeta^{\scalebox{1.0}{$\scriptscriptstyle -1$}} (u
\mathbf{x}^{\alpha\scalebox{.9}{$\scriptscriptstyle '$}})^{\scalebox{1.0}{$\scriptscriptstyle -1$}\slash n}}
= R_{\sss 2,\,1}\!\left(\ux;u\xx^{k\dd}\eval\right)
\, \cdot \, \frac{1}{1-x_{\sss 4}^{\sss 2}}
\prod_{j=1}^{3} \frac{1}{(1-x_{\sss 4}x_{\sss j})(1-x_{\sss 4}\slash x_{\sss j})}
\]
where the first factor in the right-hand side becomes
$R_{\sss 2}(u^{-2\slash n}\slash \zeta^{2})$ when evaluated
at $\ux=\u1$. Here $R_{\sss 2,\,1}(\ux;u)$ and $R_{\sss 2}(\vr)$
are the same functions as in Lemma~\ref{lem_even}.
\end{lem}
\begin{proof} The assertions follow at once from the cases
$n=1$, $\zeta=1$ and $n=2$, $\zeta=1$ of the two
aforementioned lemmas, after making the substitution
\[
u\mapsto  u\xx^{k\dd}\eval=\begin{cases}
\zeta^{{\scalebox{1.1}{$\scriptscriptstyle 1$}}-n}  (u \mathbf{x}^{\alpha\scalebox{.9}{$\scriptscriptstyle '$}})^{\scalebox{1.1}{$\scriptscriptstyle 1$}\slash n} & \text{if $n$ odd} \\
\zeta^{{\scalebox{1.1}{$\scriptscriptstyle 2$}}-n}
x_{\sss 4}^{{\scalebox{1.1}{$\scriptscriptstyle 1-2$}}\slash n}
u^{{\scalebox{1.1}{$\scriptscriptstyle 2$}} \slash n}   & \text{if $n$ even}.
\end{cases}
\]
Note that $R(\ux;u)$ is even in $u$, and so the factors
$\zeta^{n}=\pm 1$ can be ignored in the above substitution.
\end{proof}
We now give simpler formulas for the functions inside
brackets in Proposition~\ref{p9.12}, in which the dependence
upon $n$ appears only in the cocycle matrix
\tL_{\tau^{-\lfloor (n-1)\slash 2 \rfloor}}
We express the arguments in terms of
\vr=
$u^{{\scalebox{1.1}{$\scriptscriptstyle -2 $}}\slash n}$,
choosing the square root
\vr^{\scalebox{1.1}{$\scriptscriptstyle  1\slash 2$}}
:=u^{{\scalebox{1.1}{$\scriptscriptstyle -1$}}\slash n}
\vskip5pt
\begin{cor} \label{C9.14} --- Let $\a\in \Phi_{n}$,
$w_{\a}\in W$ and $R_{n}(\vr)$ be as in Lemma~\ref{lem_odd},
for $n$ odd, or as in Lemma~\ref{lem_even}, for $n$\linebreak
even. \!Then:

a). If $n=2k+1$, we have
\[
R_{n}(\vr)\ov{f_{w_{\a}}}(\u1, \vr^{\sss 1\slash 2} ; \vr^{\sss -n\slash 2})=
R_{\sss 1}(\vr) \tL_{\tau^{-k}}(\u1, \vr^{\sss 1\slash 2} ; \vr^{\sss -n\slash 2 })\cdot v_{\sss 0}
\]
where we recall that $R_{\sss 1}(\vr)=\qpoc{\vr}{\vr}^{\sss -11}$.

b). If $n=2k+2$, we have
\[
R_{n}(\vr)\ov{f_{w_{\a}}}(\u1, \vr^{\sss 1\slash 2} ; \vr^{\sss -n\slash 2 })=
R_{\sss 2}(\vr) \tL_{\tau^{-k}}(\u1, \vr^{\sss 1\slash 2} ; \vr^{\sss -n\slash 2 })\cdot
\ov{f_{\ss_{\scalebox{.67}{$\scriptscriptstyle 4$} } t}}(\u1, \vr^{\sss 1\slash 2} ; \vr^{\sss -1 })
\]
where we recall that
R_{\sss 2}(\vr)=\qpoc{\vr}{\vr}^{\sss -8}\qpoc{\vr}{\vr^{\sss
2}}^{\sss -6}(1-1\slash \vr)^{\sss -7}
We also have
\[
\ov{f_{\ss_{\scalebox{.67}{$\scriptscriptstyle 4$} } t}}(\u1,
\vr^{\sss 1\slash 2} ; \vr^{\sss -1 })= \, \!
^{^t}\!\!\left(\tfrac{\vr^{4} + 7 \vr^{3} + 13 \vr^{2} + 7 \vr + 1}{\vr^{4}},
\tfrac{\vr^{3} + 7 \vr^{2} + 7 \vr + 1}{\vr^{7 \slash 2}},
\tfrac{3 \vr^{2} + 7 \vr + 3}{\vr^{3}}\right)\!.
\]
\end{cor}
\begin{proof} The identities follow by expressing
\vC_{\alpha,\, 1}(\underline{x};u)
in two ways, using~\eqref{49} and Lemma~\ref{L9.13}. After
can-\linebreak celling the singular parts, one can then evaluate at
$\ux=\u1$ by using Lemma~\ref{lem_odd}
(resp. Lemma~\ref{lem_even}) for $n$ odd\linebreak (resp. \!$n$ even),
and Lemma~\ref{L9.14}.
\end{proof}
We are now ready to show the following:
\vskip5pt
\begin{prop} \label{p9.14} --- For $n\ge 1$, put
\vr=q^{\scalebox{1.1}{$\scriptscriptstyle -1\slash$}n}<1
Then the leading coefficient of $Q_{n}(D,q)$ is
\[
(-1)^{\lfloor n \slash 2 \rfloor } \vr^{3  \lfloor n\slash 2 \rfloor \cdot  \lfloor (n+1)\slash 2 \rfloor   }
\cdot g_{\sss n,D}(\sqrt{\vr})
\]
where $g_{\sss n, D}\ne 0$ is a power series with non-negative
coefficients depending only upon $\lfloor D\slash 2 \rfloor$
modulo $n$.
\end{prop}
\begin{proof} \!As before, put
k=\lfloor (n-1) \slash 2 \rfloor
When $k=0$ (i.e., $n = 1, 2$), our assertion can be verified
directly from\linebreak Proposition~\ref{p9.12} and
Corollary~\ref{C9.14}; the cocycle $\tL_{\tau^{-k}}$ is the
identity matrix in this case. If $k>0$, we first observe that the
entries of the matrix
\tL_{\tau^{-k}}(\u1, \vr^{\sss 1\slash 2} ; \vr^{\sss -n\slash 2 })
in Corollary~\ref{C9.14} are of the form
times non-zero power series in~$\vr^{\sss 1\slash 2} $
with non-negative coefficients. \!Indeed, by the cocycle relation,
we can write
%\begin{equation} \label{eq: cocyle-pos}
\[\tL_{\tau^{-k}}( \underline{1}, x_{\sss 5} ; 1\slash x_{\sss 5}^{\sss n})=
\tB(\underline{1},  x_{\sss 5} ; 1\slash x_{\sss 5}^{\sss n})
\tB(\underline{1},  x_{\sss 5} ; 1\slash x_{\sss 5}^{\sss n-2})
\cdot \ldots\cdot \tB(\underline{1},  x_{\sss 5} ; 1\slash x_{\sss 5}^{\sss n-2k+2})
\]%\end{equation}
and note that $n-2k+2\ge 3$. Now, for $i\ge 3$, Lemma~\ref{LA.1}
gives the functional equation
\begin{equation} \label{eq: f-eq-positivity}
\tB(\underline{1},  x_{\sss 5} ; 1\slash x_{\sss 5}^{\sss i})
=-\, x_{\sss 5}^{\sss 6(i-1)}\cdot
\, \!^t\!\tB(\underline{1},  x_{\sss 5} ; x_{\sss 5}^{\sss i-2}).
\end{equation}
By Theorem~\ref{Key-ingredient}, the function
$\tB(\u1, x_{\sss 5}; u)$ has poles only if
$ux_{\sss 5}^{j}=\pm 1$ ($j=0,1,2$), and so both sides of
\eqref{eq: f-eq-positivity}\linebreak are defined.
\!Furthermore, by Proposition~\ref{pa.2}, the entries of the matrix in the
right-hand side of \eqref{eq: f-eq-positivity} are non-zero power
series in $x_{\sss 5}$ with positive coefficients, and thus the
assertion about
\tL_{\tau^{-k}}(\u1, \vr^{\sss 1\slash 2} ; \vr^{\sss -n\slash 2 })
fol-\linebreak lows at once by taking $x_{\sss 5}=\vr^{\sss 1\slash 2}$ and
summing the exponents
% $3(n-1), 3(n-3), \ldots,3( n-2k+1)$
of $\vr$ coming from \eqref{eq: f-eq-positivity}.

The remaining factors in the right-hand sides of the formulas
in Corollary~\ref{C9.14} are also power series with non-negative
coefficients, except for the factor
\[
(1-1\slash \vr)^{\sss -7} = - \vr^{\sss 7}
\left(1 + \vr + \vr^{\sss 2} + \vr^{\sss 3} + \cdots \right)^{\sss 7}
\]
in the formula of $R_{\sss 2}(\vr)$. \!Thus, when $n$ is even,
the product
- \vr^{\sss 7}
\ov{f_{\ss_{\scalebox{.67}{$\scriptscriptstyle 4$} } t}}(\u1,
\vr^{\sss 1\slash 2} ; \vr^{\sss -1 })
gives an additional factor of $(-\vr)^{\sss 3}$. Finally, to see
that the series obtained after applying Proposition~\ref{p9.12}
is not identically zero, one\linebreak just needs to notice that the
power series of $R_{\sss 1}(\vr)$ and $R_{\sss 2}(\vr)$ contain
powers $\vr^{\sss j}$ with non-zero coefficients for all residue
classes $j$ modulo $n$ (coming for example from the expansion
of $(1-\vr)^{\sss -1}$). This completes the proof.
\end{proof}

\begin{rem} \!The estimate of the power of $\vr$ dividing the leading
term in the previous proposition is essentially optimal.
To see this, let us assume that $D$ is even and that
D\slash 2\equiv 3 \lfloor n\slash 2 \rfloor \cdot  \lfloor
(n+1)\slash 2 \rfloor \pmod n
Then the leading coefficient of the polynomial $Q_{n}(D,q)$,
expanded as a power series in
\vr^{\sss 1\slash 2} = q^{\sss -1\slash 2n}
is asymptotically
\[
(-1)^{\lfloor n \slash 2 \rfloor } \vr^{3  \lfloor n\slash 2 \rfloor \cdot  \lfloor (n+1)\slash 2 \rfloor }
(1+O(\vr)) \cdot\begin{cases}
\frac{1}{2^{\scalebox{.85}{$\scriptscriptstyle 4$}}
n^{\scalebox{.85}{$\scriptscriptstyle 10$}}} \prod_{j=0}^{3}
\frac{(2j)!}{(4+j)!} & \text{ if $n$ odd}\\
\frac{1}{7!n^{\scalebox{.85}{$\scriptscriptstyle 7$}}} & \text{ if $n$ even}.
\end{cases}
\]
Indeed, by Lemma~\ref{L5.3}, one has $\tB(\xx;0)=\tB_{\sss 0}$, and
the formula follows at once from the argument of the proof of the
previous proposition, combined with Proposition~\ref{p9.12} and the
formulas at the beginning of this subsection.
\end{rem}

\appendix
\section{Properties of the matrix $B(\xx; u)$}
We collect here two properties of the matrix $B(\xx;u)$ in
Theorem~\ref{Key-ingredient} that were needed in the proofs
of Corollary~\ref{C5.5} and Proposition~\ref{p9.14}.

From the explicit formula of the matrix
B(\xx;u)=A^{-1}(\xx; u\xx^{\dd})
with $A$ given in~[1], one verifies the follow-\linebreak ing
functional equation.
\vskip5pt
\begin{lem}\label{LA.1} --- The matrix
$\tB(\xx;u)=B(\xx;u)\slash (1-u^{\sss 2}x^\dd)^{\sss 2}$ satisfies
\[
^t\!\tB\big(\xx; 1\slash u\xx^\dd\big)=-u^{\sss 6}\xx^{{\sss 3}\dd} \tB(\xx;u).
\]
\end{lem}

The main result of this appendix is the following:
\vskip5pt
\begin{prop}\label{pa.2} --- Each entry of
$\tB(\underline{1},x_{\sss 5};u)$ is of the form
\[
\sum_{\substack{a,b,c\ge 0 \\a+b+c=11}}\frac{p_{\sss a,b,c}(x_{\sss 5},u)}
{\left(1-u^{\sss 2}\right)^{\scalebox{1.1}{$\scriptscriptstyle a$}}
\!\left(1-u^{\sss 2}x_{\sss 5}^{\sss
2}\right)^{\scalebox{1.1}{$\scriptscriptstyle b$}}
\!\left(1-u^{\sss 2}x_{\sss 5}^{\sss
4}\right)^{\scalebox{1.1}{$\scriptscriptstyle c$}}} \ne 0
\]
where $p_{\sss a,b,c}(x,y)$ are polynomials with non-negative coefficients.
\end{prop}
Let $\vr=x_{\sss 5}^{\sss 2}$, $q=u^{\sss 2}$, and set $r=\vr q$.
We have:
\[
\tB(\underline{1},x_{\sss 5};u)= \begin{pmatrix}
\frac{b_{\scalebox{.7}{$\scriptscriptstyle
4$}}(r-1)^{\scalebox{.85}{$\scriptscriptstyle 11$}}
(\vr r-1)^{\scalebox{.85}{$\scriptscriptstyle 4$}}}   &  \frac{\sqrt{r}\,
4$}}(r-1)^{\scalebox{.85}{$\scriptscriptstyle 11$}} }   &
\frac{\sqrt{q}\,b_{\scalebox{.7}{$\scriptscriptstyle
4$}}(r-1)^{\scalebox{.85}{$\scriptscriptstyle 10$}} (\vr
r-1)^{\scalebox{.85}{$\scriptscriptstyle 4$}}}    \\[8pt]
\frac{\sqrt{r}\, b_{\scalebox{.7}{$\scriptscriptstyle 21$}}}{(\vr
4$}}(r-1)^{\scalebox{.85}{$\scriptscriptstyle 11$}} } &
\frac{b_{\scalebox{.7}{$\scriptscriptstyle
22$}}}{(r-1)^{\scalebox{.85}{$\scriptscriptstyle 11$}}} &
\frac{\sqrt{\vr}\, b_{\scalebox{.7}{$\scriptscriptstyle 23$}}}{(\vr
4$}}(r-1)^{\scalebox{.85}{$\scriptscriptstyle 10$}} } \\[8pt]
\frac{\sqrt{q}\,b_{\scalebox{.7}{$\scriptscriptstyle
4$}}(r-1)^{\scalebox{.85}{$\scriptscriptstyle 10$}} (\vr
r-1)^{\scalebox{.85}{$\scriptscriptstyle 4$}}}  &\frac{\sqrt{\vr}\,
4$}}(r-1)^{\scalebox{.85}{$\scriptscriptstyle 10$}} }  &
\frac{b_{\scalebox{.7}{$\scriptscriptstyle
4$}}(r-1)^{\scalebox{.85}{$\scriptscriptstyle 9$}} (\vr
r-1)^{\scalebox{.85}{$\scriptscriptstyle 4$}}}                                \end{pmatrix}
\]
where the numerators $b_{\sss ij}=b_{\sss ij}(\vr,q)$ are explicit
non-zero polynomials, given in~[1]. The proposition
fol-\linebreak lows immediately from the following decompositions
of $b_{\sss ij}$.
\vskip5pt
\begin{lem}\label{LA.2} --- We have
b_{\sss 22}(\vr,q)= p_{\sss
where the polynomial $p_{\sss 22}$ has negative coefficients,
and the other numerators decompose as follows
\[
b_{\sss 11}=\vec{a}_{\sss 8}\cdot \vec{p}_{\sss 11},\quad
b_{\sss 12}=\vec{e}_{\sss 4} \cdot\vec{p}_{\sss 12},\quad
b_{\sss 21}=\vec{f}_{\sss 4} \cdot\vec{p}_{\sss 21},
\]
\[
b_{\sss 13}=(q-1)\vr \vec{a}_{\sss 6}\cdot\vec{p}_{\sss 13}+
(r-1)^{\sss 3}\vec{a}_{\sss 4}\cdot\vec{q}_{\sss 13}, \quad
b_{\sss 31}=(\vr r-1)\vec{a}_{\sss 6}\cdot\vec{p}_{\sss 13}+
(r-1)^{\sss 3}\vec{a}_{\sss 4}\cdot\vec{q}_{\sss 13},
\]
\[
b_{\sss 23}=\vec{f}_{\sss 3}\cdot\vec{p}_{\sss 23}, \quad
b_{\sss 32}=\vec{e}_{\sss 3}\cdot\vec{p}_{\sss 32}, \quad
b_{\sss 33}=\vec{a}_{\sss 6}\cdot \vec{p}_{\sss 33}
\]
in terms of the vectors
\vec{a}_{\sss 2k}=\left[(r-1)^{\sss 2k-2i}(q-1)^{\sss i}(\vr r-1)^{\sss i} \right]_{i=0, \ldots, k },$
\[
\vec{e}_{\sss k}=\left[(r-1)^{\sss k-i}(q-1)^{\sss i}\right]_{i=0, \ldots, k },\quad
\vec{f}_{\sss k}=\left[(r-1)^{\sss k-i}(\vr r-1)^{\sss i}\right]_{i=0, \ldots, k }
\]
where $\vec{p}_{\sss ij}$ and $\vec{q}_{\sss 13}$ denote vectors
of non-zero polynomials with negative coefficients
of the same size as the first vector in the scalar products.
\end{lem}
\begin{proof} The decompositions can be verified using the explicit
formulas of the vectors $\vec{p}_{ij}$ and $\vec{q}_{\sss 13}$
in~[1]. By Lemma~\ref{LA.1}, the polynomials
$b_{ji}(\vr,q)$ can be expressed in terms of
$b_{ij}(\vr,1/q\vr^{\sss 2})$, and so we only have to consider half
of the off-diagonal entries of $\tB$. We illustrate how one
arrives at these decompositions by considering an example.

For the entry $b_{\sss 33}$, we notice that
b_{\sss 33}(\vr,1/\vr^{n})=
(\vr-1)^{\sss 6}g_{n}(\vr)
for $n=0,1,2,\ldots$, where $g_{n}$ are Laurent poly-\linebreak
nomials with negative coefficients. Therefore we look for a
decomposition involving the factors in the de-\linebreak
nominator of the $(3,3)$ entry of the form:
\[
b_{\sss 33}(\vr,q)= \sum_{i=0}^3 (r-1)^{\sss 6-2i}(q-1)^{\sss i}
(\vr r-1)^{\sss i} h_{i}(r)
\]
with $h_{i}(r)$ polynomials in $r=\vr q$. (This assumption on
$h_{i}$ and the type of decomposition is suggested by the
symmetry of the diagonal entries under
$q\mapsto 1\slash q \vr^{\sss 2}$.) Under this assumption,
the polynomials $h_{i}$ can be determined recursively,
by letting $h_{\sss 0}(r)=b_{\sss 33}(r,1)\slash (r-1)^{\sss 6}$,
and repeating the same procedure with $b_{\sss 33}(\vr,q)$ replaced
\[
\frac{b_{\sss 33}(\vr,q)-b_{\sss 33}(r,1)}{(q-1)(\vr r-1)}
\]
etc. All the coefficients of the polynomials $h_{\sss 0}(r)$
and $h_{\sss 3}(r)$ thus found are already negative, and only
two\linebreak of the coefficients of $h_{\sss 1}(r)$ and $h_{\sss 2}(r)$
are positive. \!To eliminate the latter, we replace $h_{\sss 1}(r)$ and
$h_{\sss 2}(r)$ in the above decomposition by
\[
h_{\sss 2}'(r)=h_{\sss 2}(r)-(r-1)^{\sss 2}(r+r^{\sss 7}), \quad
h_{\sss 1}'(\vr,q)=h_{\sss 1}(r)+(r+r^{\sss 7})(q-1)(\vr r-1)
\]
so that $h_{\sss 2}'(r)$ has now all coefficients negative.
\!In the same way, we replace $h_{\sss 1}'(\vr,q)$ and
$h_{\sss 0}(r)$ by $h_{\sss 1}''(\vr,q)$\linebreak and $h_{\sss
0}'(\vr,q)$, respectively, so that $h_{\sss 1}''(\vr,q)$ has only negative
coefficients, and find that $h_{\sss 0}'(\vr,q)$ has only\linebreak
negative coefficients as well.

The same method can be used to decompose the entries
$b_{\sss 11}$, $b_{\sss 12}$ and $b_{\sss 32}$ in terms of the
entries of the corresponding vector $\vec{a}_{\sss 2k}$ or
$\vec{e}_{\sss k}$, as in the statement of the lemma. For
$b_{\sss 13}$, we decompose both
\[
(b_{\sss 13}(\vr,q)\pm b_{\sss 31}(\vr,q))\slash (\vr\pm 1)
\]
in terms of the entries of $\vec{a}_{\sss 6}$, with coefficients
that are again polynomials of $r$ alone. We recover
$b_{\sss 13}(\vr,q)$ as a linear combination of these decompositions.
\end{proof}

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