2.1 Primitive cubic characters and $L$ -functions

In the non-Kummer setting, primitive cubic characters over $\mathbb {F}_q[T]$ are most conveniently constructed by restricting cubic characters over $\mathbb {F}_{q^2}[T]$ (see [Reference Bary-Soroker and Meisner2]).

Assume $q\equiv 2\ {\mathrm{mod}}\ 3$ and fix an isomorphism $\Omega $ between the cubic roots of unity in $\mathbb {C}^\ast $ and those in $\mathbb {F}_{q^2}^\ast $ . Let $\pi \in \mathbb {F}_{q^2}[T]$ be monic irreducible such that $\pi \pi ^\sigma =P$ is irreducible in $\mathbb {F}_q[T]$ , where $\sigma $ denotes the nontrivial element of $\mathrm {Gal}(\mathbb {F}_{q^2}/\mathbb {F}_q)$ . For $a\in \mathbb {F}_{q^2}[T]$ , define the cubic residue symbol $\chi _\pi (a)$ by

$$\begin{align*}\chi_\pi(a)= \begin{cases} 0, & \pi\mid a,\\ \alpha, & \pi\nmid a, \end{cases} \end{align*}$$

where $\alpha $ is the unique cubic root of unity in $\mathbb {C}$ such that

$$ \begin{align*} a^{\frac{q^{2\deg\pi}-1}{3}}\equiv\Omega(\alpha)\quad {\mathrm{mod}}\ \pi. \end{align*} $$

Then $\chi _\pi |_{\mathbb {F}_q[T]}$ is a cubic character modulo $P$ .

We extend this definition by multiplicativity. For a monic polynomial $F=\pi _1^{e_1}\dots \pi _s^{e_s}\in \mathbb {F}_{q^2}[T]$ with distinct $\pi _i$ , set $\chi _F=\chi _{\pi _1}^{e_1}\dots \chi _{\pi _s}^{e_s}$ . Restricting $\chi _F$ to $\mathbb {F}_q[T]$ yields a cubic character on $\mathbb {F}_q[T]$ . Moreover, $\chi _F$ is primitive if and only if $e_i=1$ for all $i$ and $F$ has no prime divisor in $\mathbb {F}_q[T]$ ; in this case, the conductor is $FF^\sigma $ .

Let ${\mathcal {A}_{q}}$ (resp. ${\mathcal {A}_{q^2}}$ ) denote the set of monic polynomials over $\mathbb {F}_q$ (resp. $\mathbb {F}_{q^2}$ ), and let $\mathcal {H}_q$ (resp. ${\mathcal {H}_{q^2}}$ ) denote the set of monic square-free polynomials in ${\mathcal {A}_{q}}$ (resp. ${\mathcal {A}_{q^2}}$ ). Recall that ${\mathcal {H}_{q^2, g/2+1}}$ denotes the set of monic square-free polynomials in ${\mathcal {A}_{q^2}}$ of degree $g/2+1$ . We write $|a|=q^{\deg a}$ for $a\in {\mathcal {A}_{q}}$ and $|a|_2=q^{2\deg a}$ for $a\in {\mathcal {A}_{q^2}}$ . With this notation, David et al. [Reference David, Florea and Lalín6] show the following lemma.

Lemma 2.1 (Lemma 2.11 in [Reference David, Florea and Lalín6])

Suppose $q\equiv 2\ {\mathrm{mod}}\ 3$ . Then

(2.1) $$ \begin{align} \sum_{\substack{\chi\ \mathrm{primitive}\ \mathrm{cubic}\\ \mathrm{genus}(\chi)=g}}L_q\left(\frac{1}{2}, \chi\right) =\sum_{\substack{F\in{\mathcal{H}_{q^2, g/2+1}} \\ P|F\Rightarrow P\not\in \mathbb{F}_q[T]}}L_q\left(\frac{1}{2}, \chi_{F}\right). \end{align} $$

Since

$$ \begin{align*} \chi_F(\alpha)=\Omega^{-1}\left(\alpha^{\frac{q^{2\deg F}-1}{3}}\right) \end{align*} $$

for $\alpha \in \mathbb {F}_q\subset \mathbb {F}_{q^2}$ , and $q$ is odd with $q\equiv 2\ {\mathrm{mod}}\ 3$ , every cubic character over $\mathbb {F}_q[T]$ is even, that is, $\chi |_{\mathbb {F}_q^\ast }=1$ . Consequently, we have the following functional equation.

Lemma 2.2 (Functional equation for even $L$ -functions)

Let $F\in {\mathcal {H}_{q^2}}$ and suppose that $F$ has no prime divisor in ${\mathcal {A}_{q}}$ . Then

$$\begin{align*}L_q(s,\chi_{F}) =\epsilon(\chi_F)q^{2s-1}\frac{1-q^{-s}}{1-q^{s-1}}\frac{L_q(1-s,\overline{\chi_F})}{|F|_2^{s-\frac{1}{2}}}, \end{align*}$$

where $\epsilon (\chi _F)=q^{-\deg F}G(\chi _F)$ and $G(\chi _F)$ is the Gauss sum defined in Definition 2.1.

This is Lemma 2.3 in [Reference David, Florea and Lalín6].

Next, we record bounds for cubic $L$ -functions; this restates Lemmas 2.6 and 2.7 of [Reference David, Florea and Lalín6].

Lemma 2.3 (Lindelöf Hypothesis)

Let $\chi $ be a primitive cubic character of conductor $h\not =1$ defined over $\mathbb {F}_q[T]$ . Then, for $\Re (s)\ge \frac {1}{2}$ and all $\varepsilon>0$ ,

(2.2) $$ \begin{align} |L_q(s,\chi)|\ll q^{\varepsilon\deg h}; \end{align} $$

and for $\Re (s)\ge 1$ and all $\varepsilon>0$ ,

(2.3) $$ \begin{align} |L_q(s,\chi)|\gg q^{-\varepsilon\deg h}. \end{align} $$

2.2 Gauss sums

We recall standard properties of polynomial Gauss sums (see, for instance, [Reference Zheng13]).

Definition 2.1 For $\chi $ a primitive character of modulus $h$ on $\mathbb {F}_q[T]$ , define

$$\begin{align*}G(\chi)=\sum_{a \quad{\mathrm{mod}}\ h}\chi(a)e_q\left(\frac{a}{h}\right). \end{align*}$$

Here,

$$ \begin{align*} e_q(a/h)=e^{2\pi i\text{Tr}_{\mathbb{F}_q/\mathbb{F}_p}(a_1)/p}, \end{align*} $$

where $a_1$ is the coefficient of $1/T$ in the Laurent expansion of $\frac {a}{h}$ in $\mathbb {F}_q((1/T))$ .

We also define generalized Gauss sums as follows.

Definition 2.2 Let $\chi _f$ be a cubic character of $\mathbb {F}_q[T]$ defined as above for some ${f\in \mathbb {F}_q[T]}$ . We define

(2.4) $$ \begin{align} G_q(V, f)=\sum_{u\quad {\mathrm{mod}}\ f}\chi_f(u)e_q\left(\frac{uV}{f}\right). \end{align} $$

This is Lemma 2.12 of [Reference David, Florea and Lalín6]. For the primitive character $\chi _F$ defined above, we have the following lemma.

Lemma 2.5 For $F\in \mathbb {F}_{q^2}[T]$ , we have

$$\begin{align*}G(\chi_F)=G_{q^2}(1,F). \end{align*}$$

Moreover, if $F$ is square-free, then

(2.6) $$ \begin{align} |G_{q^2}(1,F)|=q^{\deg F}. \end{align} $$

2.3 Distribution of Gauss sums

The generating series for Gauss sums is defined by

(2.7) $$ \begin{align} \Psi_q(f, u)=\sum_{F\in{\mathcal{A}_{q}}}G_q(f,F)u^{\deg F}, \end{align} $$

and

(2.8) $$ \begin{align} \widetilde{\Psi}_q(f,u)=\sum_{\substack{F\in{\mathcal{A}_{q}}\\ (F,f)=1}}G_q(f,F)u^{\deg F}. \end{align} $$

We now discuss the distribution of Gauss sums. Patterson established the distribution of cubic Gauss sums for $S$ -integers (see [Reference Patterson10]); with minor adjustments, we obtain the following estimate.

Lemma 2.6 (Lemma 3.11 in [Reference David, Florea and Lalín6])

If $q^{-\frac {1}{2}}\ge |u|\ge q^{-\frac {3}{2}}$ and both $|u^3-q^{-4}|>\delta $ and ${|u^3-q^{-2}|>\delta }$ , then

$$\begin{align*}\widetilde{\Psi}_q(f,u)\ll_{\delta} |f|^{\frac{3}{4}+\varepsilon}|u^{\frac{1}{2}}|^{\deg f}. \end{align*}$$

2.4 Multivariable complex analysis lemmas

Our approach uses two standard results from multivariable complex analysis. We begin with the notion of a tube domain.

Definition 2.3 An open set $T\subset \mathbb {C}^n$ is a tube if there exists an open set $U\subset \mathbb {R}^n$ such that

$$\begin{align*}T=\{z\in\mathbb{C}^n:\ \Re(z)\in U\}. \end{align*}$$

For any set $U\subset \mathbb {C}^n$ , we define $T(U)=U+i\mathbb {R}^n\subset \mathbb {C}^n$ . We quote Bochner’s Tube Theorem [Reference Bochner3].

We denote the convex hull of an open set $T\subset \mathbb {C}^n$ by $\widehat T$ . Our next result is [Reference Čech4, Proposition C.5] on the preservation of modulus inequalities under holomorphic continuation.

3.1 The generating function

Now, we construct a double Dirichlet series to rewrite the sum in equation (2.1). From now on, we will always use $P$ , $P_1$ , or $P_2$ to denote irreducible polynomial in corresponding polynomial ring. For $\Re (s)$ and $\Re (w)$ sufficiently large, we define

(3.1) $$ \begin{align} A_3(s, w)=\sum_{\substack{F\in\mathcal{H}_{q^2} \\ P|F\Rightarrow P\not\in \mathbb{F}_q[T]}} \frac{L_q(w, \chi_{F})}{|F|^s_2}. \end{align} $$

Upon taking $u=q^{-2s}$ , we rewrite the sum as

(3.2) $$ \begin{align} \sum_{\substack{F\in{\mathcal{H}_{q^2, g/2+1}} \\ P|F\Rightarrow P\not\in \mathbb{F}_q[T]}}L_q\left(\frac{1}{2}, \chi_{F}\right) &= \frac{1}{2\pi i}\oint_{|u|=r}\sum_{\substack{F\in\mathcal{H}_{q^2} \\ P|F\Rightarrow P\not\in \mathbb{F}_q[T]}} L_q\left(\frac{1}{2}, \chi_{F}\right) u^{\deg F}\frac{du}{u^{g/2+2}}\notag\\&=\frac{1}{2\pi i}\oint_{|u|=r}A_3\left(u,\frac{1}{2}\right)\frac{du}{u^{g/2+2}}, \end{align} $$

by applying Perron’s formula for function fields

(3.3) $$ \begin{align} \sum_{n\le N}a(n)=\frac{1}{2\pi i}\oint_{|u|=r}\left(\sum_{n=0}^{\infty}a(n)u^n\right)\frac{du}{(1-u)u^{N+1}}. \end{align} $$

Here, $A_3(u, w)=A_3(s, w)$ . Hereafter, we will use $A_3(s,w)$ and $A_3(u,w)$ interchangeably when there is no ambiguity.

Next, we consider the different expressions of $A_3(s,w)$ on different convergence regions and find out the convex hull of convergence regions.

3.2 The first convergence region

To analyze the behavior of $A_3(u,v)$ , we begin by using Möbius inversion

(3.4) $$ \begin{align} \sum_{\substack{D\in{\mathcal{A}_{q}}\\ D|F}}\mu(D)=\left\{ \begin{array}{@{}ll} 1 & F \text{ has no prime divisor in } \mathbb{F}_q[T]; \\ 0 & \text{otherwise}. \end{array} \right. \end{align} $$

This allows us to remove the divisor condition. Applying this, $A_3(s,w)$ can be rewritten as

(3.5) $$ \begin{align} A_3(s, w)=&\sum_{\substack{F\in{\mathcal{H}_{q^2}} \\ P|F\to P\not\in \mathbb{F}_q[T]}}\frac{L_q(w, \chi_{F})}{|F|_2^s}\notag\\ =&\sum_{D\in{\mathcal{A}_{q}}} \mu(D)\sum_{\substack{F\in\mathcal{H}_{q^2}\\ (D,F)=1}} \frac{L_q(w, \chi_{DF})}{|DF|_2^s}\notag\\ =&\sum_{D\in{\mathcal{A}_{q}}}\mu(D)\sum_{\substack{F\in\mathcal{H}_{q^2}\\ (D,F)=1}}\sum_{N\in{\mathcal{A}_{q}}}\frac{\chi_{DF}(N)}{|N|^{w}|DF|_2^s}\notag\\ =&\sum_{N\in{\mathcal{A}_{q}}}\frac{1}{|N|^w}\sum_{\substack{D\in{\mathcal{A}_{q}}\\ }}\frac{\mu(D)\chi_D(N)}{|D|_2^s}\sum_{\substack{F\in{\mathcal{H}_{q^2}}\\ (D,F)=1}}\frac{\chi_F(N)}{|F|_2^s}\notag\\ =&\sum_{N\in{\mathcal{A}_{q}}}\frac{1}{|N|^w}\frac{L_{q^2}(s, \chi^{(N)})}{L_{q^2}(2s, \bar{\chi}^{(N)})}\sum_{\substack{D\in{\mathcal{A}_{q}}\\ }}\frac{\mu(D)\chi_D(N)}{|D|_2^s}\prod_{\substack{P_2|D\\ P_2\in{\mathcal{A}_{q^2}}}}\left(1+\frac{\chi_{P_2}(N)}{|P_2|_2^s}\right)^{-1}. \end{align} $$

The last equality uses a trick

$$ \begin{align*} \sum_{\substack{F\in{\mathcal{H}_{q^2}}\\ (D,F)=1}}\frac{\chi_F(N)}{|F|_2^s}=\prod_{\substack{P_2\in{\mathcal{A}_{q^2}}\\ (P_2,D)=1}}\left(1+\frac{\chi_{P_2}(N)}{|P_2|^s_2}\right)=\frac{L_{q^2}(s,\chi^{(N)})}{L_{q^2}(2s,\overline{\chi}^{(N)})}\prod_{\substack{P_2\in{\mathcal{A}_{q^2}}\\ P_2|D}}\left(1+\frac{\chi_{P_2}(N)}{|P_2|_2^s}\right)^{-1}. \end{align*} $$

Here, $\chi ^{(N)}(F)=\chi _F(N)$ is the Hecke character induced by $\chi _F$ . $L_{q^2}(s,\chi ^{(N)})$ denotes the $L$ -function over $\mathbb {F}_{q^2}(T)$ . Meanwhile, we recall that $|F|=q^{\deg F}$ and $|F|_2=q^{2\deg F}$ .

Using the lemma below, we simplify $A_3(s,w)$ further.

Proof Let $\sigma : \mathbb {F}_{q^2}\to \mathbb {F}_{q^2}$ be the nontrivial Frobenius automorphism. It’s sufficient to treat the case $D=P\in {\mathcal {A}_{q}}$ is irreducible. Assume first that $P$ does not split in extension $\mathbb {F}_{q^2}(T)/\mathbb {F}_{q}(T)$ . By definition, we have

$$ \begin{align*}F^{\frac{q^{2\deg P}-1}{3}}\equiv \Omega(\alpha)\quad {\mathrm{mod}}\ P,\end{align*} $$

which is equivalent to say $F^{\frac {q^{2\deg P}-1}{3}}=f(T)P+\Omega (\alpha ).$ Applying $\sigma $ and using ${\sigma (F)=F}$ , we have $F^{\frac {q^{2\deg P}-1}{3}}=f^{\sigma }(T)P+\sigma (\Omega (\alpha ))\equiv \sigma (\Omega (\alpha ))\ {\mathrm{mod}}\ P$ . Since two constants congruent modulo $P$ must coincide, $\sigma (\Omega (\alpha ))= \Omega (\alpha )$ , hence $\Omega (\alpha )=1$ . If instead ${P=\pi \pi ^{\sigma }}$ splits, then $\chi _{P}(F)=\chi _{\pi }\overline {\chi _{\pi }}(F)=1$ . For the general case, $D=P_1^{e_1}\dots P_k^{e_k}$ follows by multiplicativity, since $\chi _D=\chi _{P_1}^{e_1}\dots \chi _{P_k}^{e_k}$ .

Using the lemma, we obtain

$$ \begin{align*} A_3(s, w)=&\sum_{N\in{\mathcal{A}_{q}}}\frac{1}{|N|^{w}}\frac{L_{q^2}(s, \chi^{(N)})}{L_{q^2}(2s, \bar{\chi}^{(N)})}\sum_{\substack{D\in{\mathcal{A}_{q}}\\ (D,N)=1}}\frac{\mu(D)}{|D|^s_2}\prod_{\substack{P_2\in{\mathcal{A}_{q^2}}\\ P_2|D}}\left(1+\frac{\chi_{P_2}(N)}{|P_2|^s_2}\right)^{-1}\\ =&\sum_{N\in{\mathcal{A}_{q}}}\frac{1}{|N|^{w}}\frac{L_{q^2}(s, \chi^{(N)})}{L_{q^2}(2s, \bar{\chi}^{(N)})}\prod_{\substack{P_1\in{\mathcal{A}_{q}}\\ (P_1,N)=1}}\left(1-\frac{1}{|P_1|^s_2}\prod_{\substack{P_2|P_1\\ P_2\in{\mathcal{A}_{q^2}}}}\left(1+\frac{\chi_{P_2}(N)}{|P_2|_2^s}\right)^{-1}\right) \\ =&\sum_{N\in{\mathcal{A}_{q}}}\frac{1}{|N|^{w}}\frac{L_{q^2}(s, \chi^{(N)})}{L_{q^2}(2s, \bar{\chi}^{(N)})}P\left(s,\chi^{(N)}\right)\prod_{\substack{P_1|N\\ P_1\in{\mathcal{A}_{q}}}}\left(1-\frac{1}{|P_1|_2^s}\right)^{-1}, \end{align*} $$

where

(3.6) $$ \begin{align} P(s, \chi^{(N)})=\prod_{P_1\in{\mathcal{A}_{q}}}\left(1-\frac{1}{|P_1|_2^s}\prod_{\substack{P_2|P_1\\ P_2\in{\mathcal{A}_{q^2}}}}\left(1+\frac{\chi_{P_2}(N)}{|P_2|_2^s}\right)^{-1}\right). \end{align} $$

For $\Re (s)>\frac {1}{2}$ , it follows that

(3.7) $$ \begin{align} P(s,\chi^{(N)})&=\prod_{P_1\in{\mathcal{A}_{q}}}\left(1-\frac{1}{|P_1|_2^s}+O\left(\frac{1}{|P_1|_2^{3s/2}}\right)\right)\notag\\&=\zeta_q^{-1}(2s)\prod\limits_{P\in{\mathcal{A}_{q}}}\left(1+O\left(\frac{1}{|P|^{3s/2}_2}\right)\right). \end{align} $$

As $(1-\frac {1}{|P_1|_2^s})^{-1}=1+\frac {1}{|P_1|_2^s-1}\ll |P_1|_2^{\max \{0,-\Re (s)\}}$ , we will obtain

(3.8) $$ \begin{align} \prod_{\substack{P_1|N\\ P_1\in{\mathcal{A}_{q}}}}\left(1-\frac{1}{|P_1|_2^s}\right)^{-1}\ll |N|_2^{\max\{0,-\Re(s)\}+\varepsilon}. \end{align} $$

Taking $u=q^{-2s}$ and $v=q^{-w}$ , we have

(3.9) $$ \begin{align} A_3(u,v)=\sum_{N\in{\mathcal{A}_{q}}}v^{\deg N}\frac{L_{q^2}(u, \chi^{(N)})}{L_{q^2}(u^2, \bar{\chi}^{(N)})}P\left(u,\chi^{(N)}\right)\prod_{\substack{P_1|N\\ P_1\in{\mathcal{A}_{q}}}}\left(1-u^{\deg P_1}\right)^{-1}, \end{align} $$

where

$$ \begin{align*}L_{q^2}\left(u,\chi^{(N)}\right)=\sum\limits_{F\in{\mathcal{A}_{q^2}}}\chi_F(N)u^{\deg F}\end{align*} $$

and

$$ \begin{align*}P\left(u, \chi^{(N)}\right)=\prod\limits_{P_1\in{\mathcal{A}_{q}}}\left(1-u^{\deg P_1}\prod\limits_{\substack{P_2\in{\mathcal{A}_{q^2}}\\ P_2|P_1}}\left(1+\chi_{P_2}(N)u^{\deg P_2}\right)^{-1}\right).\end{align*} $$

It is easy to see that $P(u, \chi ^{(N)})$ is analytic for $|u|<q^{-1}$ .

The function $L_{q^2}(u,\chi ^{(N)})$ is a polynomial in $u$ and convergent for $|u|<q^{-1}$ in general. If $N$ is a cube, then $L_{q^2}(u,\chi ^{(N)})$ has a simple pole at $u=q^{-2}$ and $L_{q^2}(u^2,\chi ^{(N)})^{-1}$ is a polynomial in $u$ . If $N$ is not a cube, then $L_{q^2}(u,\chi ^{(N)})$ is analytic and $L_{q^2}(u^2,\chi ^{(N)})$ has zeros on the circle $|u|=q^{-1/2}$ . Hence, $(u-q^{-2})\frac {L_{q^2}(u,\chi ^{(N)})}{L_{q^2}(u^2,\chi ^{(N)})}$ is analytic for $|u|<q^{-\frac {1}{2}}$ .

When $N$ is not a cube, $\frac {L_{q^2}(u, \chi ^{(N)})}{L_{q^2}(u^2, \bar {\chi }^{(N)})}$ is a rational function in $u$ with possible simple poles on the line $|u|=q^{-\frac {1}{2}}$ . The Lindelöf Hypothesis gives, for $|u|<q^{-1}$ and arbitrary $\varepsilon>0$ ,

(3.10) $$ \begin{align} \frac{L_{q^2}(u, \chi^{(N)})}{L_{q^2}(u^2, \bar{\chi}^{(N)})}\ll |N|_2^{\varepsilon}.\end{align} $$

There is a possible simple pole at $u=q^{-2}$ when $N$ is a cube in $A_3(u,v)$ . In fact, combining (3.7)–(3.10), for $|u|<q^{-1}$ and any $\varepsilon>0$ , we have the bound

(3.11) $$ \begin{align} |(u-q^{-2})A_3(u,v)|\ll |(u-q^{-2})(1-qu)|\sum_{N\in{\mathcal{A}_{q}}}|q^{\varepsilon}v|^{\deg N}\ll |(u-q^{-2})(1-qu)|\zeta_{q}(vq^{\varepsilon}). \end{align} $$

Meanwhile, $(u-q^{-2})A_3(u,v)$ is holomorphic in the convergence region of the right-hand side of the above inequality:

(3.12) $$ \begin{align} S_1=\{(u,v)||u|<q^{-1}, |v|<q^{-1}\}. \end{align} $$

Remark 3.1 In $(s,w)$ -coordinates, the region $S_1$ becomes

(3.13) $$ \begin{align} S_1=\{(s,w)|\Re(s)>\frac{1}{2}, \Re(w)>1\}. \end{align} $$

We will always describe all convergence regions in the $(s,w)$ -coordinate, since Bochner’s Tube Theorem is stated for tube domains.

3.3 Residue at $u=q^{-2}$

We now analyze the possible poles at $u=q^{-2}$ . Let $\zeta _{q^2}(u)=\frac {1}{1-q^2u}$ be the zeta function for the field $\mathbb {F}_{q^2}(T)$ . The possible pole comes from the term $N$ being a cube. We have

(3.14) $$ \begin{align} &\mathrm{Res}_{u=q^{-2}}A_3(u,v)\notag\\&\quad =\mathrm{Res}_{u=q^{-2}}\sum_{N\in{\mathcal{A}_{q}}}v^{3\deg N}\frac{\zeta_{q^2}(u)}{\zeta_{q^2}(u^2)}\prod_{\substack{P_2\in{\mathcal{A}_{q^2}}\\P_2|N}}\frac{1-u^{\deg P_2}}{1-u^{2\deg P_2}}\prod_{\substack{P_1\in{\mathcal{A}_{q}}\\ P_1 \nmid N}}\left(1-u^{\deg P_1}\prod_{\substack{P_2\in{\mathcal{A}_{q^2}}\\ P_2|P_1}}(1+u^{\deg P_2})^{-1}\right)\notag\\&\quad =P(q^{-2})Z(q^{-2},v)\mathrm{Res}_{u=q^{-2}}\frac{\zeta_{q^2}(u)}{\zeta_{q^2}(u^2)},\notag\\ \end{align} $$

where

(3.15) $$ \begin{align} P(u)=\prod_{P_1\in{\mathcal{A}_{q}}}\left(1-u^{\deg P_1}\prod_{\substack{P_2\in{\mathcal{A}_{q^2}}\\ P_2|P_1}}(1+u^{\deg P_2})^{-1}\right) \end{align} $$

and

(3.16) $$ \begin{align} Z(u,v)=\sum_{N\in{\mathcal{A}_{q}}}v^{3\deg N}\prod_{\substack{P_2\in{\mathcal{A}_{q^2}}\\P_2|N}}(1+u^{\deg P_2})^{-1}\prod_{\substack{P_1\in{\mathcal{A}_{q}}\\ P_1|N}}\left(1-u^{\deg P_1}\prod_{\substack{P_2\in{\mathcal{A}_{q^2}}\\ P_2|P_1}}(1+u^{\deg P_2})^{-1}\right)^{-1}. \end{align} $$

Taking $v=q^{-\frac {1}{2}}$ , we have

$$ \begin{align*} \mathrm{Res}_{u=q^{-2}}A_3(u,q^{-\frac{1}{2}})=(q^{-4}-q^{-2})P(q^{-2})Z(q^{-2},q^{-\frac{1}{2}}). \end{align*} $$

3.4 The second region

The Lindelöf Hypothesis gives, for $|v|\le q^{-\frac {1}{2}}$ and any $\varepsilon>0$ ,

$$ \begin{align*}|L_q(v,\chi_F)|\ll |F|^{\varepsilon}\end{align*} $$

and hence

$$ \begin{align*} {\sum_{\substack{F\in\mathcal{H}_{q^2} \\ P|F\Rightarrow P\not\in \mathbb{F}_q[T]\\ F\not= 1}}L_q(v,\chi_F)u^{\deg F}\ll \sum_{F\in\mathcal{H}_{q^2}}|F|^{\varepsilon}|u|^{\deg F}\ll\frac{\zeta_{q^2}(uq^{\varepsilon})}{\zeta_{q^2}((uq^{\varepsilon})^2)}.} \end{align*} $$

For the term $F=1$ , we note that

$$ \begin{align*} (v-q^{-1})\zeta_q(v)\ll 1 \end{align*} $$

for all $v$ , where $\zeta _q(v)=\frac {1}{1-qv}$ is the zeta function of $\mathbb {F}_q(T)$ . Therefore, $(v-q^{-1}) A_3(u,v)$ is holomorphic in the convergence region

(3.17) $$ \begin{align} S_{2,1}=\{(u,v)|~|v|\le q^{-\frac{1}{2}},|u|<q^{-2}\}. \end{align} $$

Similarly, when $|v|\ge q^{-\frac {1}{2}}$ , the Lindelöf Hypothesis with the functional equation yields, for any $\varepsilon>0$ ,

$$ \begin{align*} {|L_q(v,\chi_F)|\ll |v^2q^{1+\varepsilon}|^{\deg F}.} \end{align*} $$

Consequently,

$$ \begin{align*} {A_3(u,v)=\zeta_{q}(v)+\sum_{\substack{F\in\mathcal{H}_{q^2} \\ P|F\Rightarrow P\not\in \mathbb{F}_q[T]\\ F\not=1}}L_q(v,\chi_F)u^{\deg F}\ll \sum_{F\in\mathcal{H}_{q^2}} |v^2uq^{1+\varepsilon}|^{\deg F}\ll\frac{\zeta_{q^2}(uv^2q^{(1+\varepsilon)})}{\zeta_{q^2}((uv^2q^{(1+\varepsilon)})^2)}} \end{align*} $$

in the convergence region

(3.18) $$ \begin{align} {S_{2,2}=\{(u,v)|~|v|\ge q^{-\frac{1}{2}},|u^{1/2}v|<q^{-\frac{3}{2}}\}.} \end{align} $$

Combining the two cases, we obtain

(3.19) $$ \begin{align} S_2=S_{2,1}\cup S_{2,2}=\{(u,v)|~|u^{1/2}v|<q^{-\frac{3}{2}}, |u|<q^{-2}\}. \end{align} $$

Remark 3.2 In $(s,w)$ -coordinates, this region becomes

(3.20) $$ \begin{align} S_2=\{(s,w)|\Re(s+w)>\frac{3}{2}, \Re(s)>1\}. \end{align} $$

4.1 The third convergence region

We now derive the third convergence region. The functional equation (Lemma 2.2) gives

(4.1) $$ \begin{align} A_3(s,w)&=\sum_{\substack{F\in\mathcal{H}_{q^2} \\ P|F\Rightarrow P\not\in \mathbb{F}_q[T]}} \frac{L_q(w, \chi_{F})}{|F|^s_2}\notag\\& =q^{2w-1}\frac{1-q^{-w}}{1-q^{w-1}}\sum_{\substack{F\in\mathcal{H}_{q^2} \\ P|F\Rightarrow P\not\in \mathbb{F}_q[T]}}\frac{L_q(1-w, \overline{\chi}_F)}{|F|_2^{s+w}}G_{q^2}(1,F)\notag\\& =q^{2w-1}\frac{1-q^{-w}}{1-q^{w-1}}\sum_{N\in{\mathcal{A}_{q}}}\frac{1}{|N|^{1-w}}\sum_{D\in{\mathcal{A}_{q}}}\frac{\mu(D)\overline{\chi}_D(N)G_{q^2}(1,D)}{|D|_2^{s+w}}\sum_{\substack{F\in{\mathcal{H}_{q^2}}\\ (F,DN)=1}}\frac{G_{q^2}(ND,F)}{|F|_2^{s+w}}. \end{align} $$

Replacing $q$ by $q^2$ , Lemma 2.4 shows $G_{q^2}(ND,F)=0$ whenever $F$ is not square-free. Hence, we may rewrite $A_3(s,w)$ as

$$ \begin{align*} A_3(s,w) &=q^{2w-1}\frac{1-q^{-w}}{1-q^{w-1}}\sum_{N\in{\mathcal{A}_{q}}}\frac{1}{|N|^{1-w}}\sum_{\substack{D\in{\mathcal{A}_{q}}\\ (D,N)=1}}\frac{\mu(D)G_{q^2}(N,D)}{|D|_2^{s+w}}\sum_{\substack{F\in{\mathcal{A}_{q^2}}\\ (F,DN)=1}}\frac{G_{q^2}(ND,F)}{|F|_2^{s+w}}\\&=q^{2w-1}\frac{1-q^{-w}}{1-q^{w-1}}C_3(s,w), \end{align*} $$

where

(4.2) $$ \begin{align} C_3(s,w)=\sum_{N\in{\mathcal{A}_{q}}}\frac{1}{|N|^{1-w}}\sum_{\substack{D\in{\mathcal{A}_{q}}\\ (D,N)=1}}\frac{\mu(D)G_{q^2}(N,D)}{|D|_2^{s+w}}H(ND,s+w) \end{align} $$

and

(4.3) $$ \begin{align} H(Q,s)=\sum_{\substack{F\in{\mathcal{A}_{q^2}}\\ (F,Q)=1}}\frac{G_{q^2}(Q,F)}{|F|_2^s}. \end{align} $$

Taking $u=q^{-2s}$ and $v=q^{-w}$ , $A_3(u,v)$ becomes

(4.4) $$ \begin{align} A_3(u,v)=\frac{1}{qv^2}\frac{1-v}{1-\frac{1}{qv}} C_3(u,v), \end{align} $$

where

(4.5) $$ \begin{align} C_3(u,v)=\sum_{N\in{\mathcal{A}_{q}}}\frac{1}{(qv)^{\deg N}}\sum_{\substack{D\in{\mathcal{A}_{q}}\\ (D,N)=1}}\mu(D)G_{q^2}(N,D)H(ND,uv^2)(uv^2)^{\deg D} \end{align} $$

and

(4.6) $$ \begin{align} H(Q,u)=\sum_{\substack{F\in{\mathcal{A}_{q^2}}\\ (F,Q)=1}}G_{q^2}(Q,F)u^{\deg F}. \end{align} $$

As before, we do not distinguish $C_3(s,w)$ from $C_3(u,v)$ nor $H(Q,s)$ from $H(Q,u)$ when no confusion can arise.

Replacing $q$ by $q^2$ in Lemma 2.6, we obtain $H(ND,uv^2)\ll |u^{\frac {1}{2}}vq^{\frac {3}{2}+\varepsilon }|^{\deg ND}$ for $q^{-\frac {3}{2}}\le |u^{\frac {1}{2}}v|\le q^{-\frac {1}{2}}$ . Moreover,

$$ \begin{align*} A_3(u,v)&\ll\frac{1}{qv^2}\frac{1-v}{1-\frac{1}{qv}} \sum_{N\in{\mathcal{A}_{q}}}\frac{1}{|qv|^{\deg N}}\sum_{\substack{D\in{\mathcal{A}_{q}}\\ (D,N)=1}}|G_{q^2}(N,D)||uv^2|^{\deg D}|u^{\frac{1}{2}}vq^{\frac{3}{2}+\varepsilon}|^{\deg ND}\\ &\ll \frac{1}{qv^2}\frac{1-v}{1-\frac{1}{qv}}\sum_{N\in{\mathcal{A}_{q}}}\frac{1}{|qv|^{\deg N}}\sum_{\substack{D\in{\mathcal{A}_{q}}\\ (D,N)=1}}|D||uv^2|^{\deg D}|u^{\frac{1}{2}}vq^{\frac{3}{2}+\varepsilon}|^{\deg ND}\\ &\ll \frac{1}{qv^2}\frac{1-v}{1-\frac{1}{qv}}\sum_{N\in{\mathcal{A}_{q}}} |u^{\frac{1}{2}}q^{\frac{1}{2}+\varepsilon}|^{\deg N}\sum_{\substack{D\in{\mathcal{A}_{q}}\\ (D,N)=1}} |q^{\frac{5}{2}+\varepsilon}u^{\frac{3}{2}}v^3|^{\deg D}\\ &\ll \frac{1}{qv^2}\frac{1-v}{1-\frac{1}{qv}}\zeta_q(u^{\frac{1}{2}}q^{\frac{1}{2}+\varepsilon})\zeta_q(u^{\frac{3}{2}}v^3q^{\frac{5}{2}+\varepsilon}). \end{align*} $$

It follows that $A_3(u,v)$ is holomorphic in region

$$ \begin{align*}S_3=\{|u|<q^{-3},|u^{\frac{1}{2}}v|<q^{-\frac{7}{6}}, |v|>1, q^{-\frac{3}{2}}\le |u^{\frac{1}{2}}v|\le q^{-\frac{1}{2}}\}.\end{align*} $$

Remark 4.1 In $(s,w)$ -coordinates, $S_3$ becomes

(4.7) $$ \begin{align} S_3=\{(s,w)|\Re(s)>\frac{3}{2}, \Re(s+w)>\frac{7}{6}, \frac{1}{2}\le \Re(s+w)\le \frac{3}{2}, \Re(w)<0\}. \end{align} $$

We now combine the three domains obtained above. Recall that $S_1$ and $S_2$ were first obtained in $(u,v)$ -coordinates and then translated into $(s,w)$ -coordinates (see Remarks 3.1 and 3.2), while $S_3$ is already stated in $(s,w)$ -coordinates. Since Bochner’s Tube Theorem applies to tube domains in $\mathbb {C}^2$ , we work in the $(s,w)$ -system and compute the convex hull of $S_1\cup S_2\cup S_3$ . The convex hull in $(s,w)$ -coordinates can be determined by elementary geometric computation, and we obtain

(4.8) $$ \begin{align} S_4=\{(s,w)|\Re(s)>\frac{1}{2},\ \Re(s+w)>\frac{7}{6},\ \Re(4s+3w)>5\}. \end{align} $$

By Bochner’s Tube Theorem, $(u-q^{-2})(v-q^{-1})A_3(u,v)$ admits a holomorphic continuation to the corresponding region in $(u,v)$ -coordinates, namely,

(4.9) $$ \begin{align} S_4=\{(u,v)||u|<q^{-1},\ |u^{\frac{1}{2}}v|<q^{-\frac{7}{6}},\ |u^2v^3|<q^{-5}\}. \end{align} $$

4.2 Complete the proof

We now complete the proof by evaluating the integral (3.2). Fix $v=q^{-\frac {1}{2}}$ . We shift the contour from $|u|=r$ to $|u|=q^{-(\frac {7}{4}+\varepsilon )}$ . The only pole crossed is the simple pole at $u=q^{-2}$ . Therefore,

$$ \begin{align*} \frac{1}{2\pi i}\oint_{|u|=r}A_3(u,\frac{1}{2})\frac{du}{u^{g/2+2}} =\mathrm{Res}_{u=q^{-2}}\frac{A_3(u,\frac{1}{2})}{u^{g/2+2}} +\frac{1}{2\pi i}\oint_{|u|=q^{-(\frac{7}{4}+\varepsilon)}}A_3(u,\frac{1}{2})\frac{du}{u^{g/2+2}}. \end{align*} $$

The residue at $u=q^{-2}$ produces the main term in (1.4). For the shifted contour integral, since $A_3(u,v)$ is holomorphic on the contour $|u|=q^{-(\frac {7}{4}+\varepsilon )}$ , we bound it by

$$ \begin{align*} \ll q^{\frac{g}{2}(\frac{7}{4}+\varepsilon)} \oint_{|u|=q^{-(\frac{7}{4}+\varepsilon)}}|A_3(u,\frac{1}{2})|\,|du| \ll q^{(\frac{7}{8}+\varepsilon)g}, \end{align*} $$

which yields the claimed error term in (1.4).