The integral moments and ratios of quadratic Dirichlet L-functions over monic irreducible polynomials in $\mathbb {F}_{q}[T]$

Abstract

In this paper, we extend to the function field setting the heuristics formerly developed by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments of L-functions. We also adapt to the function field setting the heuristics first developed by Conrey, Farmer and Zirnbauer to the study of mean values of ratios of L-functions. Specifically, the focus of this paper is on the family of quadratic Dirichlet L-functions $L(s,\chi _{P})$ where the character $\chi$ is defined by the Legendre symbol for polynomials in $\mathbb {F}_{q}[T]$ with $\mathbb {F}_{q}$ a finite field of odd cardinality, and the averages are taken over all monic and irreducible polynomials P of a given odd degree. As an application, we also compute the formula for the one-level density for the zeros of these L-functions.

1 Introduction

In this paper, we study the family of quadratic Dirichlet L-functions $L(s,\chi _{P})$ where the character $\chi$ is defined by the Legendre symbol $\left( \frac{\cdot }{P}\right)$ and P ranges over monic irreducible polynomials of degree $2g+1$ over $\mathbb {F}_{q}[T]$. We present the conjectures for the moments and ratios of this family of L-functions, Conjectures 2.2 and 2.3, respectively, by making use of the recipe developed by Conrey et al. in [7] and adapting it for this family of L-functions.

The study of moments of families of L-functions is a central topic in analytic number theory. Many mathematicians have studied this subject, and considerable progress was made in the last decades in the direction of getting a better understanding of the asymptotic behaviour of such moments. For example, in the case of the Riemann zeta-function, the problem is to understand the asymptotic behaviour of

$$\begin{aligned} M_k(T)=\int _{0}^{T} \left| \zeta \left( \tfrac{1}{2}+it\right) \right| ^{2k}\mathrm{{d}}t \end{aligned}$$

(1.1)

as $T\rightarrow \infty$.

Hardy and Littlewood [22] proved in 1918 an asymptotic formula for the second moment, i.e.

$$\begin{aligned} M_1(T)\sim T\log T. \end{aligned}$$

(1.2)

In 1926, Ingham [24] showed that when $k=2$,

$$\begin{aligned} M_2(T)\sim \frac{1}{2\pi ^2} T \left( \log T\right) ^4. \end{aligned}$$

(1.3)

For values of $k\ge 3$, it still remains an unsolved problem to obtain asymptotic formulas for $M_{k}(T)$. However, it is conjectured that for every $k \ge 0$, there is a constant $c_k$ such that

$$\begin{aligned} M_k(T)\sim c_k T \left( \log T\right) ^{k^2}. \end{aligned}$$

(1.4)

Conrey and Ghosh [13] made a conjecture for the sixth moment of the Riemann zeta-function, and later on, Conrey and Gonek [14] put forward a conjecture for the eighth moment but their approach fails to provide conjectures for higher moments. Keating and Snaith [28], using random matrix theory, conjectured the precise value of the constant $c_k$ for all values of $k>0$. In fact, their conjecture produces a value for $c_{k}$ for $\mathfrak {R}(k)>-1/2$. More recently, Conrey and Keating, in a series of papers [9,10,11,12], returned to the problem of obtaining conjectures for the higher moments of the Riemann zeta-function using only number-theoretic heuristics. Their new approach produced conjectures for moments of the Riemann zeta function, as well as explained the role of the non-diagonal contribution to the main terms in the asymptotic formulas.

A different example is the family of quadratic Dirichlet L-functions $L(s,\chi _d)$, where $\chi _d$ is the real primitive Dirichlet character modulo d defined by the Kronecker symbol $\chi _d(n)=\left( \frac{d}{n}\right)$. The problem here is to establish an asymptotic formula for

$$\begin{aligned} \sum _{d\le X} L\left( \tfrac{1}{2},\chi _d\right) ^k \end{aligned}$$

(1.5)

as $X\rightarrow \infty$, where the sum is taken over all positive discriminants d and k is a positive integer. In this case, as it is for the Riemann zeta-function, just the first few moments were computed. In 1981, Jutila [25] established the asymptotic formula for the first and second moments. The asymptotic formulas he obtained are

$$\begin{aligned} \sum _{d\le X} L\left( \tfrac{1}{2},\chi _d\right) \sim C_1 X \log X \end{aligned}$$

(1.6)

and

$$\begin{aligned} \sum _{d\le X} L\left( \tfrac{1}{2},\chi _d\right) ^2\sim C_2 X \left( \log X\right) ^3, \end{aligned}$$

(1.7)

where the constants $C_1$ and $C_2$ can be expressed in terms of Euler products and factors containing the Riemann zeta-function. Soundararajan [32] computed the asymptotic formula for the third moment. He proved that

$$\begin{aligned} \sum _{d\le X} L\left( \tfrac{1}{2},\chi _{8d} \right) ^3\sim C_3 X \left( \log X\right) ^6, \end{aligned}$$

(1.8)

where d is an odd, square-free and positive number, $\chi _{8d}$ is a real, even primitive Dirichlet character with conductor 8d, and $C_3$ is a constant.

In another paper, Soundararajan and Young [33] claimed that they are able to establish an asymptotic formula for the fourth power moment for this family of L-functions under the Generalised Riemann Hypothesis (GRH). The claim is that

$$\begin{aligned} \sum _{d\le X} L\left( \tfrac{1}{2},\chi _{d} \right) ^4\sim C_4 X \left( \log X\right) ^{10}, \end{aligned}$$

(1.9)

where $C_4$ is constant. Recently, Shen [31] proved the asymptotic formula for the fourth moment of quadratic Dirichlet L-functions under the Generalised Riemann Hypothesis (GRH). He consider the characters of the form $\chi _{8d}$ and has established that

$$\begin{aligned} \underset{\begin{array}{c} d\le X \\ (d,2)=1 \end{array}}{\sum \nolimits ^{*}} L\left( \tfrac{1}{2},\chi _{8d}\right) ^4 \sim \frac{a_4}{2^6\cdot 3^3\cdot 5^2 \cdot 7 \cdot \pi } X(\log X)^{10}, \end{aligned}$$

(1.10)

where $a_4$ is as defined in [27].

In 2005, Conrey et al. [7] presented a new heuristic for all of the main terms in the integral moments of several families of primitive L-functions. Their conjectures agree with previously known results. For the Riemann zeta-function, they gave a precise conjecture for $M_k(T)$ including an asymptotic expansion for the lower-order terms using shifted moments. For the family of quadratic Dirichlet L-functions, they conjectured that

$$\begin{aligned} \sum _{d} L\left( \tfrac{1}{2},\chi _d\right) ^k=\sum _{d} Q_k (\log |d|)(1+o(1)), \end{aligned}$$

(1.11)

where $Q_k$ is polynomial of degree $k(k+1)/2$ with $k\in {\mathbb N}$.

It is important to observe that Diaconu et al. [16] have also conjectured moments of families of L-functions using different techniques. Their method is based on multiple Dirichlet series. Recently, Diaconu and Whitehead [17] established a smoothed asymptotic formula for the third moment of quadratic Dirichlet L-functions at the central value. In addition to the main term, which is known, they prove the existence of a secondary term of size $x^{3/4}$. The error term in their asymptotic formula is on the order of $O(x^{2/3+\delta })$ for every $\delta >0$.

In 2008, Conrey et al. [8] presented a generalisation of the heuristic method for moments presented in [7] to the case of ratios of products of L-functions. These conjectures are very powerful since they encode information about statistics of zeros of such L-functions. The ratios conjectures as put forward by Conrey, Farmer and Zirnbauer can be used to prove very precise results about the distribution of zeros of families of L-functions such as pair-correlation and n-level density (for more details see [15]). Their ratios conjecture for the family of quadratic Dirichlet L-functions are read as follow.

Conjecture 1.1

(Conrey, Farmer, Zirnbauer) Let $\mathcal {D}=\{L(s,\chi _d):d>0\}$ to be the symplectic family of L-functions associated with the quadratic character $\chi _d$. For positive real parts of $\alpha _k$ and $\gamma _m$, we have

$$\begin{aligned} \begin{aligned}&\sum _{0<d\le X} \frac{\prod _{k=1}^K L\left( \frac{1}{2}+\alpha _k,\chi _d\right) }{\prod _{m=1}^Q L\left( \frac{1}{2}+\gamma _m,\chi _d\right) }\\&\quad = \sum _{0<d\le X} \sum _{\varepsilon \in \{-1,1\}^K}\left( \frac{|d|}{\pi }\right) ^{\frac{1}{2}\sum _{k=1}^K \left( \varepsilon _k\alpha _k-\alpha _k\right) }\\&\quad \quad \times \prod _{k=1}^K g_+\left( \tfrac{1}{2}+\tfrac{\alpha _k-\varepsilon _k\alpha _k}{2}\right) Y\left( \varepsilon _1\alpha _1,\ldots ,\varepsilon _K\alpha _K;\gamma \right) A_\mathcal {D}\left( \varepsilon _1\alpha _1,\ldots ,\varepsilon _K\alpha _K;\gamma \right) \\&\quad \quad + o(X), \end{aligned} \end{aligned}$$

(1.12)

where

$$\begin{aligned} g_+(s)&= \frac{\Gamma \left( \frac{1-s}{2}\right) }{\Gamma \left( \frac{s}{2}\right) }, \end{aligned}$$

(1.13)

$$\begin{aligned} Y(\alpha ;\gamma )&= \frac{\prod _{j\le k\le K} \zeta \left( 1+\alpha _j+\alpha _k\right) \prod _{m\le r\le Q} \zeta \left( 1+\gamma _m+\gamma _r\right) }{\prod _{k=1}^K\prod _{m=1}^Q \zeta \left( 1+\alpha _k+\gamma _m\right) } \end{aligned}$$

(1.14)

and

$$\begin{aligned} \begin{aligned} A_\mathcal {D} (\alpha ;\gamma )&=\prod _{\begin{array}{c} p \end{array}} \frac{\prod _{j\le k\le K} \left( 1-\frac{1}{p^{1+\alpha _j+\alpha _k}}\right) \prod _{m\le r\le Q} \left( 1-\frac{1}{p^{1+\gamma _m+\gamma _r}}\right) }{\prod _{k=1}^K\prod _{m=1}^Q \left( 1-\frac{1}{p^{1+\alpha _k+\gamma _m}}\right) }\\&\quad \times \left( 1+ \left( 1+\frac{1}{p}\right) ^{-1} \sum _{0< \sum _ka_k+\sum _mc_m \text { is even}} \frac{\prod _{m=1}^Q \mu \left( P^{c_m}\right) }{p^{\sum _ka_k(\frac{1}{2}+\alpha _k)+\sum _m c_m(\frac{1}{2}+\gamma _m)}}\right) . \end{aligned} \end{aligned}$$

(1.15)

In 1979, Goldfeld and Viola [21] introduced a variant of the problem about moments of quadratic Dirichlet L-functions. They conjectured an asymptotic formula for

$$\begin{aligned} \sum _{\begin{array}{c} p\le X \\ p \equiv 3 \text { (mod} 4) \end{array}} L\left( \tfrac{1}{2},\chi _p\right) , \end{aligned}$$

(1.16)

where the sum is taken over prime numbers and $\chi _p(n)= (\tfrac{n}{p})$ is the usual Legendre symbol. In this direction, Jutila [25] studied the first moment of this family of L-functions and proved that

$$\begin{aligned} \sum _{\begin{array}{c} p\le X \\ p \equiv 3 \text { (mod} 4) \end{array}} \left( \log p\right) L\left( \tfrac{1}{2},\chi _p\right) \sim \frac{1}{4} X \log X. \end{aligned}$$

(1.17)

Recently, assuming the Generalised Riemann Hypothesis (GRH), Baluyot and Pratt [5] obtained the leading order term for the second moment. They proved that

$$\begin{aligned} \sum _{\begin{array}{c} p\le X \\ p \equiv 1 \text { (mod} 8) \end{array}} \left( \log p\right) L\left( \tfrac{1}{2},\chi _p\right) ^2 = c \frac{X}{4}\left( \log X\right) ^3 + O \left( X\left( \log X\right) ^{11/4}\right) , \end{aligned}$$

where c is a positive constant.

We should notice that the second moment for this family of L-functions seems to be the limit of the current technology. This is in part due to the fact that for this family, we are dealing with character sums over prime numbers, and these sums are more complicated than those over square-free numbers. For example, in the case for square-free numbers, it was possible to obtain the third moment by making use of the Poisson summation formula, but the same does not seem to apply for the family over prime numbers since we cannot directly apply Poisson to the sums over primes.

1.1 The function field setting

Let $\mathcal {H}_{2g+1,q}$ be the hyperelliptic ensemble of monic, square-free polynomials of degree $2g+1.$ When the cardinality of the field $\mathbb {F}_q$ is $q\equiv 1 \pmod 4$, Andrade and Keating [2] computed the first moment of the family of L-functions associated to the quadratic character $\chi _{D}$, with $D\in \mathcal {H}_{2g+1,q}.$ They proved that

$$\begin{aligned} \sum _{D\in \mathcal {H}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _D\right) \sim |D|P_1\left( \log _q|D|\right) , \end{aligned}$$

(1.18)

where $P_1$ is a linear polynomial. For the second, third and fourth moments of this family, Florea [18, 19] proved that

$$\begin{aligned} \sum _{D\in \mathcal {H}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _D\right) ^k \sim |D| P_k\left( \log _q|D|\right) , \end{aligned}$$

(1.19)

where $P_k$ is a polynomial of degree 3, 6 and 10 respectively, whose coefficients can be computed explicitly, except for $P_4$ where only the first few coefficients were obtained. It is worth noticing that Florea in [20] improved the error term for the first moment and was able to obtain a strenuous lower-order term that was never predicted by random matrix theory or other heuristics.

In another paper, Andrade and Keating [4] adapted the recipe of [7] and of [8] to the function field setting and conjectured asymptotic formulas for the integral moments and ratios of the family of quadratic Dirichlet L-functions in function fields. Their main conjectures are presented below.

Conjecture 1.2

(Andrade and Keating–Integral Moments Conjecture) Suppose that q odd is the fixed cardinality of the finite field $\mathbb {F}_q$ and let $\mathcal {X}_D(s)=|D|^{1/2-s}\mathcal {X}(s)$ and

$$\begin{aligned} \mathcal {X}(s)=q^{-1/2+s}. \end{aligned}$$

That is $\mathcal {X}_D(s)$ is the factor in the functional equation

$$\begin{aligned} L(s,\chi _D)=\mathcal {X}_D(s)L(1-s,\chi _D). \end{aligned}$$

(1.20)

Summing over fundamental discriminants $D \in \mathcal {H}_{2g+1,q}$, we have

$$\begin{aligned} \sum _{D \in \mathcal {H}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _D\right) ^k=\sum _{D\in \mathcal {H}_{2g+1,q}} Q_k (\log _q|D|)(1+o(1)), \end{aligned}$$

(1.21)

where $Q_k$ is polynomial of degree $k(k+1)/2$ given by the k-fold residue

$$\begin{aligned} \begin{aligned} Q_k(x) =&\frac{(-1)^{k(k-1)/2} \, 2^k}{k!} \, \frac{1}{(2\pi i )^k} \, \oint \cdots \oint \frac{G(z_1, \ldots , z_k) \triangle \left( z_1^2, \ldots , z_k^2\right) ^2}{\prod _{i=1}^k z_i^{2k-1}}\\&\times q^{\frac{x}{2}\sum _{i=1}^k z_i} \, \mathrm{{d}}z_1 \ldots z_k, \end{aligned} \end{aligned}$$

(1.22)

$\Delta \left( z_1,\ldots ,z_k\right)$ the Vandermonde determinant given by

$$\begin{aligned} \Delta \left( z_1,\ldots ,z_k\right)&= \prod {1\le i\le j\le k} (z_j-z_i), \end{aligned}$$

(1.23)

$$\begin{aligned} G(z_1, \ldots , z_k)&= A_k(\tfrac{1}{2};z_1, \ldots , z_k) \prod _{i=1}^k \mathcal {X}\left( \tfrac{1}{2}+z_i\right) ^{-\frac{1}{2}} \prod _{1 \le i \le j \le k} \zeta _A(1+z_i+z_j),\nonumber \\ \end{aligned}$$

(1.24)

and $A_k$ is the Euler product, absolutely convergent for $|\mathfrak {R}(z_i)|<\frac{1}{2},$ defined by

$$\begin{aligned} \begin{aligned} A_k(\tfrac{1}{2};z_1, \ldots , z_k)&= \prod _{\begin{array}{c} P \text { monic}\\ \text {irreducible} \end{array}} \prod _{1 \le i \le j \le k} \Bigg (1- \frac{1}{|P|^{1+z_i+z_j}}\Bigg )\\&\quad \times \left( \frac{1}{2} \left( \prod _{i=1}^k \Bigg (1-\frac{1}{|P|^{\frac{1}{2}+z_i}}\Bigg )^{-1}+ \prod _{i=1}^k \Bigg (1+\frac{1}{|P|^{\frac{1}{2}+z_i}}\Bigg )^{-1} \right) +\frac{1}{|P|}\right) \\&\quad \times \left( 1+\frac{1}{|P|}\right) ^{-1}. \end{aligned} \end{aligned}$$

(1.25)

Conjecture 1.3

(Andrade and Keating–Ratios Conjecture) Let $\alpha _{k}$ and $\gamma _{m}$ complex numbers with positive and small real parts. Let $\mathfrak {D}=\{L(s,\chi _D):D\in \mathcal {H}_{2g+1,q}\}$ to be the family of L-functions associated with the quadratic character $\chi _D$. Then,

$$\begin{aligned} \begin{aligned}&\sum _{D\in \mathcal {H}_{2g+1,q}} \frac{\prod _{k=1}^K L\left( \tfrac{1}{2}+\alpha _k,\chi _D\right) }{\prod _{m=1}^Q L\left( \tfrac{1}{2}+\gamma _m,\chi _D\right) }\\&\quad = \sum _{D\in \mathcal {H}_{2g+1,q}} \sum _{\varepsilon \in \{-1,1\}^k} \left| D\right| ^{-\frac{1}{2} \sum _{k=1}^K \left( \varepsilon _k\alpha _k-\alpha _k\right) } \prod _{k=1}^K \mathcal {X}\left( \tfrac{1}{2}+\tfrac{\alpha _k-\varepsilon _k\alpha _k}{2}\right) \\&\quad \quad \times Y_\mathfrak {D}\left( \varepsilon _1\alpha _1,\ldots ,\varepsilon _K\alpha _K;\gamma \right) A_\mathfrak {D}\left( \varepsilon _1\alpha _1,\ldots ,\varepsilon _K\alpha _K;\gamma \right) + o\left( D\right) , \end{aligned} \end{aligned}$$

(1.26)

with

$$\begin{aligned} \begin{aligned} A_\mathfrak {D} (\alpha ;\gamma )\\&\quad =\prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \frac{\prod _{j\le k\le K} \left( 1-\frac{1}{|P|^{1+\alpha _j+\alpha _k}}\right) \prod _{m\le r\le Q} \left( 1-\frac{1}{|P|^{1+\gamma _m+\gamma _r}}\right) }{\prod _{k=1}^K\prod _{m=1}^Q \left( 1-\frac{1}{|P|^{1+\alpha _k+\gamma _m}}\right) }\\&\quad \quad \times \left( 1+ \left( 1+\frac{1}{|P|}\right) ^{-1} \sum _{0< \sum _ka_k+\sum _mc_m \text { is even}} \frac{\prod _{m=1}^Q \mu \left( P^{c_m}\right) }{|P|^{\sum _ka_k(\frac{1}{2}+\alpha _k)+\sum _m c_m(\frac{1}{2}+\gamma _m)}}\right) \end{aligned} \end{aligned}$$

(1.27)

and

$$\begin{aligned} \begin{aligned} Y_\mathfrak {D}(\alpha ;\gamma )= \frac{\prod _{j\le k\le K} \zeta _A\left( 1+\alpha _j+\alpha _k\right) \prod _{m\le r\le Q} \zeta _A\left( 1+\gamma _m+\gamma _r\right) }{\prod _{k=1}^K\prod _{m=1}^Q \zeta _A\left( 1+\alpha _k+\gamma _m\right) }, \end{aligned} \end{aligned}$$

(1.28)

where $\zeta _{A}(s)$ is the zeta-function associated to the polynomial ring $A=\mathbb {F}_{q}[T]$ and $\mathcal {X}(s)$ is a function that depends on q.

One can note that (1.21) and (1.26) are the function field analogues of the formulas (1.11) and (1.12), respectively.

The main aim of this paper is to formulate a conjectural asymptotic formula for

$$\begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _P\right) ^k, \end{aligned}$$

(1.29)

where $\mathcal {P}_{2g+1,q}$ is the set of all monic, irreducible polynomials of odd degree $2g+1$ with coefficients in $\mathbb {F}_{q}$, as $|P| \rightarrow \infty$.

Andrade and Keating [3] established asymptotic formulas for the first and second moments of (1.29), namely

$$\begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \left( \log _q|P|\right) L\left( \tfrac{1}{2},\chi _P\right) \sim \frac{1}{2} |P| \left( \log _q|P|+1\right) , \end{aligned}$$

(1.30)

$$\begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _P\right) ^{2} \sim \frac{1}{24}\frac{1}{\zeta _A(2)} |P| \left( \log _q|P|\right) ^2. \end{aligned}$$

(1.31)

Recently, Bui and Florea [6] improved Andrade and Keating’s result for the second moment and proved that

$$\begin{aligned} \frac{1}{|\mathcal {P}_{2g+1,q}|} \sum _{P\in \mathcal {P}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _P\right) ^{2} = \frac{g^3}{3\zeta _A(2)}+g^2\left( \frac{3}{2}+\frac{1}{2q}\right) + O_\varepsilon \left( g^{3/2+\varepsilon }\right) .\nonumber \\ \end{aligned}$$

(1.32)

In this paper, we adapt to the function field case the recipe for the conjectures of the moments and ratios of L-functions for the family of quadratic Dirichlet L-functions associated with $\chi _P$ over a fixed finite field $\mathbb {F}_q$. In Sect. 2, we present some basic facts on L-functions over function fields followed by the statement of our main results. In Sect. 3, we present the details of the recipe of [7] when it is adapted for the function field setting. In Sect. 4, we use the integral moments conjecture over function fields when $k=1,2$, and compare with the main theorems of [4]. Then we conjecture the precise value for the third moment, i.e. when $k=3$ in this setting. In Sect. 5, we present the recipe of [8] for the same family of L-functions over function fields. In Sect. 6, we use the ratios conjecture for function fields and compute the one-level density of the zeros of this same family of L-functions.

2 Statement of the main results

In this section, we gather some basic facts about L-functions over function fields. Many of the results and notation here can also be found in [29].

Let $\mathbb {F}_q$ be a finite field of odd cardinality $q=p^a$, with p a prime. Denote the polynomial ring over $\mathbb {F}_q$ by $A=\mathbb {F}_{q}[T],$ and the rational function field by $k=\mathbb {F}_q(T).$ For a polynomial f in $\mathbb {F}_{q}[T]$, we define the norm of f by $|f|:=q^{\text {deg}(f)}.$ For $\mathfrak {R}(s)>1$, the zeta-function attached to A is defined by

$$\begin{aligned} \begin{aligned} \zeta _A(s) =\sum _{f \text { monic}} \frac{1}{|f|^s} = \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \left( 1-|P|^{-s}\right) ^{-1}. \end{aligned} \end{aligned}$$

(2.1)

Since there are $q^n$ monic polynomials of degree n, we can easily prove that

$$\begin{aligned} \zeta _A(s)=\frac{1}{1-q^{1-s}}, \end{aligned}$$

(2.2)

which provides an analytic continuation of the zeta-function to the whole complex plane, with simple pole at $s=1,$ which leads to the analogue of the Prime Number Theorem for polynomials in $A=\mathbb {F}_{q}[T]$.

Theorem 2.1

(Prime Polynomial Theorem) Let $\pi _A(n)$ denote the number of monic irreducible polynomials of degree n in A. Then

$$\begin{aligned} \pi _A(n)=\frac{q^n}{n} + O\left( \frac{q^{n/2}}{n}\right) . \end{aligned}$$

(2.3)

Now, let P be a monic irreducible polynomial, define the quadratic character $(\frac{f}{P})$ by

$$\begin{aligned} \left( \frac{f}{P}\right) = {\left\{ \begin{array}{ll} 1 &{}\quad \text { if } f \text { is a square (mod }P),P\not \mid f,\\ -1 &{}\quad \text { if } f \text { is not a square (mod }P),P\not \mid f,\\ 0 &{}\quad \text { if } P\mid f.\\ \end{array}\right. } \end{aligned}$$

(2.4)

The quadratic reciprocity law states that for A, B non-zeros and relatively prime monic polynomials, we have

$$\begin{aligned} \left( \frac{A}{B}\right) = \left( \frac{B}{A}\right) \left( -1\right) ^{\frac{q-1}{2} \text {deg}(A)\text {deg}(B)}. \end{aligned}$$

(2.5)

We denote by $\chi _P$ the quadratic character defined in terms of the quadratic residue symbol for A

$$\begin{aligned} \chi _P(f)=\left( \frac{P}{f}\right) , \end{aligned}$$

(2.6)

where $f\in A$.

In this paper, the focus will be in the family of quadratic Dirichlet L-functions associated with polynomials $P\in \mathcal {P}_{2g+1,q}$, where

$$\begin{aligned} \mathcal {P}_{2g+1,q}=\{P\in A, \text { monic, irreducible and deg}(P)=2g+1\}. \end{aligned}$$

(2.7)

The quadratic Dirichlet L-function attached to the character $\chi _P$ is defined to be

$$\begin{aligned} \begin{aligned} L\left( s,\chi _P\right)&= \sum _{\begin{array}{c} f\in A \\ f \text { monic} \end{array}} \frac{\chi _P(f)}{|f|^s} \\&= \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \left( 1-\chi _P(P)|P|^{-s}\right) ^{-1},\quad \mathfrak {R}(s)>1. \end{aligned} \end{aligned}$$

(2.8)

With the change of variables $u=q^{-s},$ $L(s,\chi _P)$ is a polynomial of degree 2g given by

$$\begin{aligned} \begin{aligned} L\left( s,\chi _P\right)&= \mathcal {L}(u,\chi _{P})= \sum _{n=0}^{2g} \sum _{\begin{array}{c} f \text { monic} \\ \text {deg}(f)=n \end{array}}\chi _P(f)u^n. \end{aligned} \end{aligned}$$

(2.9)

(see Propositions 14.6 and 17.7 in [29]).

We are now in a position to state the main conjectures of this paper.

Conjecture 2.2

Suppose that $q \equiv 1 (\text {mod }4)$ is the fixed cardinality of the finite field $\mathbb {F}_q$ and let $\mathcal {X}_P(s)=|P|^{1/2-s}\mathcal {X}(s)$ where

$$\mathcal {X}(s)=q^{-1/2+s}.$$

That is $\mathcal {X}_P(s)$ is the factor in the functional equation

$$\begin{aligned} L(s,\chi _P)=\mathcal {X}_P(s)L(1-s,\chi _p). \end{aligned}$$

(2.10)

Summing over primes $P \in \mathcal {P}_{2g+1,q}$, we have

$$\begin{aligned} \sum _{P \in \mathcal {P}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _P\right) ^k=\sum _{P\in \mathcal {P}_{2g+1,q}} Q_k (\log _q|P|)(1+o(1)), \end{aligned}$$

(2.11)

where $Q_k$ is polynomial of degree $k(k+1)/2$ given by the k-fold residue

$$\begin{aligned} \begin{aligned} Q_k(x) =\,&\frac{(-1)^{k(k-1)/2} \, 2^k}{k!} \, \frac{1}{(2\pi i )^k} \, \oint \ldots \oint \frac{G(z_1, \ldots , z_k) \triangle \left( z_1^2, \ldots , z_k^2\right) ^2}{\prod _{i=1}^k z_i^{2k-1}},\\&\times q^{\frac{x}{2}\sum _{i=1}^k z_i} \, \mathrm{{d}}z_1 \ldots z_k, \end{aligned} \end{aligned}$$

(2.12)

where $\Delta \left( z_1,\ldots ,z_k\right)$ is defined as in (1.23),

$$\begin{aligned} G(z_1, \ldots , z_k)= A_k\left( \tfrac{1}{2};z_1, \ldots , z_k\right) \prod _{i=1}^k \mathcal {X}\left( \tfrac{1}{2}+z_i\right) ^{-\frac{1}{2}} \prod _{1 \le i \le j \le k} \zeta _A(1+z_i+z_j), \end{aligned}$$

(2.13)

and $A_k$ is the Euler product, absolutely convergent for $|\mathfrak {R}(z_i)|<\frac{1}{2},$ defined by

$$\begin{aligned} \begin{aligned} A_k\left( \tfrac{1}{2};z_1, \ldots , z_k\right) =&\prod _{\begin{array}{c} P \text { monic}\\ \text {irreducible} \end{array}} \prod _{1 \le i \le j \le k} \left( 1- \frac{1}{|P|^{1+z_i+z_j}}\right) \\&\times \frac{1}{2} \left( \prod _{i=1}^k \left( 1-\frac{1}{|P|^{\frac{1}{2}+z_i}}\right) ^{-1}+ \prod _{i=1}^k \left( 1+\frac{1}{|P|^{\frac{1}{2}+z_i}}\right) ^{-1} \right) . \end{aligned} \end{aligned}$$

(2.14)

More generally, we have

$$\begin{aligned} \begin{aligned} \sum _{P \in \mathcal {P}_{2g+1,q}}&L\left( \tfrac{1}{2}+\alpha _1,\chi _P\right) \cdots L\left( \tfrac{1}{2}+\alpha _k,\chi _P\right) \\&=\sum _{P\in \mathcal {P}_{2g+1,q}} \prod _{i=1}^k \mathcal {X}\left( \tfrac{1}{2}+\alpha _i\right) ^{-\frac{1}{2}} |P|^{-\frac{1}{2}\sum _{i=1}^k\alpha _i}Q_k (\log _q|P|,\alpha )(1+o(1)) \end{aligned} \end{aligned}$$

(2.15)

where

$$\begin{aligned} \begin{aligned} Q_k(x,\alpha ) =&\frac{(-1)^{k(k-1)/2} \, 2^k}{k!} \, \frac{1}{(2\pi i )^k} \, \oint \cdots \oint \frac{G(z_1, \cdots , z_k) \triangle \left( z_1^2, \ldots , z_k^2\right) ^2 \prod _{i=1}^k z_i}{ \prod _{i=1}^k \prod _{j=1}^k (z_j-\alpha _i) (z_j+\alpha _i)}\\&\times q^{\frac{x}{2}\sum _{i=1}^k z_i} \, \mathrm{{d}}z_1 \cdots z_k, \end{aligned} \end{aligned}$$

(2.16)

and the path of integration encloses the $\pm \alpha$’s.

Note that, for the cases $k=1,2$, our conjecture agrees with Andrade and Keating’s results in (1.30) and (1.31) and Bui and Florea’s result in (1.32). See Sects. 4.1 and 4.2 for further details.

The next conjecture is the translation for function fields of the ratios conjecture for quadratic Dirichlet L-functions associated with the characters $\chi _P$.

Conjecture 2.3

Suppose that the real parts of $\alpha _k$ and $\gamma _k$ are positive and that q odd is the fixed cardinality of the finite field $\mathbb {F}_q$. Let $\mathfrak {P}=\{L(s,\chi _P):P\in \mathcal {P}_{2g+1,q}\}$ to be the family of L-functions associated with the quadratic character $\chi _P$. Then with the same notation as before, we have

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \frac{\prod _{k=1}^K L\left( \tfrac{1}{2}+\alpha _k,\chi _P\right) }{\prod _{m=1}^Q L\left( \tfrac{1}{2}+\gamma _m,\chi _P\right) }\\&\quad = \sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\varepsilon \in \{-1,1\}^k} \left| P\right| ^{-\frac{1}{2} \sum _{k=1}^K \left( \varepsilon _k\alpha _k-\alpha _k\right) } \prod _{k=1}^K \mathcal {X}\left( \tfrac{1}{2}+\tfrac{\alpha _k-\varepsilon _k\alpha _k}{2}\right) \\&\quad \quad \times Y_\mathfrak {P}\left( \varepsilon _1\alpha _1,\ldots ,\varepsilon _K\alpha _K;\gamma \right) A_\mathfrak {P}\left( \varepsilon _1\alpha _1,\ldots ,\varepsilon _K\alpha _K;\gamma \right) + o\left( P\right) , \end{aligned} \end{aligned}$$

(2.17)

where

$$\begin{aligned} \begin{aligned} A_\mathfrak {P} (\alpha ;\gamma )&=\prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \frac{\prod _{j\le k\le K} \left( 1-\frac{1}{|P|^{1+\alpha _j+\alpha _k}}\right) \prod _{m\le r\le Q} \left( 1-\frac{1}{|P|^{1+\gamma _m+\gamma _r}}\right) }{\prod _{k=1}^K\prod _{m=1}^Q \left( 1-\frac{1}{|P|^{1+\alpha _k+\gamma _m}}\right) }\\&\quad \times \left( 1+ \sum _{0< \sum _k a_k+\sum _m c_m \text { is even}} \frac{\prod _{m=1}^Q \mu \left( P^{c_m}\right) }{|P|^{\sum _ka_k(\frac{1}{2}+\alpha _k)+\sum _m c_m(\frac{1}{2}+\gamma _m)}}\right) \end{aligned} \end{aligned}$$

(2.18)

and

$$\begin{aligned} \begin{aligned} Y_\mathfrak {P}(\alpha ;\gamma )= \frac{\prod _{j\le k\le K} \zeta _A\left( 1+\alpha _j+\alpha _k\right) \prod _{m\le r\le Q} \zeta _A\left( 1+\gamma _m+\gamma _r\right) }{\prod _{k=1}^K\prod _{m=1}^Q \zeta _A\left( 1+\alpha _k+\gamma _m\right) }. \end{aligned} \end{aligned}$$

(2.19)

If we compare the above conjectures with the ones presented by Andrade and Keating in the previous section, one can immediately see that although they are similar in nature, there is an important difference between the formulas and the final shape of the conjectures. More specifically, one of the main differences is the arithmetic term that is produced in both conjectures. These factors are not the same and this is due to the fact that in one setting, we are averaging over square-free polynomials and so an Euler product is produced that needs to be carried out through the recipe and in the end produce the term $A_k$ in Andrade and Keating conjecture, while in the case presented in this paper, the average is taken over prime numbers and the final formula produces a simpler arithmetic factor due to precise formula that we have when using the prime polynomial theorem in $\mathbb {F}_{q}[T]$. This difference comes from the fourth step in the recipe when we replace each summand by its expected value.

In the following sections, we present the details of how to arrive at these conjectures.

3 Integral moments of L-functions over prime polynomials

In this section, we present the details of the recipe for conjecturing moments of the family of quadratic Dirichlet L-function $L(s,\chi _{P})$ associated to hyperelliptic curves of genus g over fixed finite field $\mathbb {F}_q$ as $g\rightarrow \infty$. As in Andrade and Keating [4], we will adjust the recipe first presented in [7] to the function field setting.

Let $P\in \mathcal {P}_{2g+1,q}.$ For a fixed k, we aim to obtain an asymptotic expression for

$$\begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _P\right) ^k \end{aligned}$$

(3.1)

as $g \rightarrow \infty .$ In order to achieve this, we consider the more general expression obtained by introducing small shifts, say $\alpha _1, \ldots , \alpha _k$

$$\begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} L\left( \tfrac{1}{2}+\alpha _1,\chi _P\right) \cdots L\left( \tfrac{1}{2}+\alpha _k,\chi _P\right) . \end{aligned}$$

(3.2)

Introducing the shifts helps to reveal the hidden structures in the form of symmetries. Moreover, the calculations are simplified by the removal of higher-order poles. In the end, letting each $\alpha _1,\cdots ,\alpha _k$ tend to 0 will provide an asymptotic formula for (3.1).

3.1 Analogies between classical L-functions and L-functions over function fields

The first step to obtaining a conjecture for the integral moments of L-functions of any family is the use of the approximate functional equation. Thus, the “approximate” functional equation for the L-function attached to the character $\chi _P$ is given by

$$\begin{aligned} L(s,\chi _P)= \sum _{\begin{array}{c} n \text { monic}\\ \text {deg}(n)\le g \end{array}} \frac{\chi _P(n)}{|n|^s} + \mathcal {X}_p(s) \sum _{\begin{array}{c} n \text { monic}\\ \text {deg}(n)\le g-1 \end{array}} \frac{\chi _P(n)}{|n|^{1-s}}, \end{aligned}$$

(3.3)

where $P\in \mathcal {P}_{2g+1,q}$ and $\mathcal {X}_P(s)=q^{g(1-2s)}.$ Note that $\mathcal {X}_P(s)$ can also be re-written as follows:

$$\begin{aligned} \mathcal {X}_P(s)=|P|^{\frac{1}{2}-s} \mathcal {X}(s), \end{aligned}$$

(3.4)

where $\mathcal {X}(s)=q^{-\frac{1}{2}+s}$ corresponds to the gamma factor that appears in the classical quadratic L-functions.

The next result, quoted from [4], makes the analogy between the function field case and the number field case more apparent.

Lemma 3.1

We have that

$$\begin{aligned} \mathcal {X}_P(s)^{\frac{1}{2}}=\mathcal {X}_P(1-s)^{-\frac{1}{2}} \end{aligned}$$

(3.5)

and

$$\begin{aligned} \mathcal {X}_P(s) \, \mathcal {X}_P(1-s)=1. \end{aligned}$$

(3.6)

Consider the following completed L-function:

$$\begin{aligned} Z_{\mathcal {L}}(s,\chi _P)=\mathcal {X}_P(s)^{-\frac{1}{2}}L(s,\chi _P). \end{aligned}$$

(3.7)

We will apply the recipe to this completed L-function, since it simplifies the calculations and satisfies a more symmetric functional equation given by the next lemma.

Lemma 3.2

Let $Z_{\mathcal {L}}(s,\chi _P)$ be the Z-function defined above, then we have the following functional equation:

$$\begin{aligned} Z_{\mathcal {L}}(s,\chi _P)=Z_{\mathcal {L}}(1-s,\chi _P). \end{aligned}$$

(3.8)

Proof

Direct from the definition of $Z_{\mathcal {L}}(s,\chi _P)$ and Lemma 3.1. $\square$

Now, let

$$\begin{aligned} L_P(s)= \sum _{P\in \mathcal {P}_{2g+1,q}} Z(s;\alpha _1, \ldots , \alpha _k) \end{aligned}$$

(3.9)

be the k-shifted moment, with

$$\begin{aligned} Z(s;\alpha _1, \ldots , \alpha _k)= \prod _{i=1}^k\, Z_{\mathcal {L}}(s+\alpha _i,\chi _P). \end{aligned}$$

(3.10)

Using the “approximate” functional equation (3.3) and Lemma 3.1, we have

$$\begin{aligned} Z_{\mathcal {L}}(s,\chi _P)= \mathcal {X}_P(s)^{-\frac{1}{2}} \sum _{\begin{array}{c} n \text { monic}\\ \text {deg}(n)\le g \end{array}} \frac{\chi _P(n)}{|n|^s} \, + \, \mathcal {X}_p(1-s)^{-\frac{1}{2}} \sum _{\begin{array}{c} n \text { monic}\\ \text {deg}(n)\le g-1 \end{array}} \frac{\chi _P(n)}{|n|^{1-s}}. \end{aligned}$$

(3.11)

3.2 Adapting the CFKRS recipe for the function field case

We present the steps of the recipe which follows from [4, 7] with the necessary modifications for the family of $L(s,\chi _{P})$.

• (1) Write the product of k-shifted L-functions.

  $$\begin{aligned} Z\left( \tfrac{1}{2};\alpha _1,\ldots , \alpha _k\right) = Z_\mathcal {L}\left( \tfrac{1}{2}+\alpha _1,\chi _P\right) \cdots Z_\mathcal {L}\left( \tfrac{1}{2}+\alpha _k,\chi _P\right) . \end{aligned}$$

  (3.12)

• (2) Replace each L-function with the two terms from its approximate functional equation (3.3) with $s=1/2+\alpha _i$.

  $$\begin{aligned} \begin{aligned} Z(\tfrac{1}{2}; \alpha _1, \ldots , \alpha _k)&= \sum _{\varepsilon _i=\pm 1} \prod _{i=1}^k \mathcal {X}_P(\tfrac{1}{2}+\varepsilon _i\alpha _i)^{-\frac{1}{2}} \sum _{\begin{array}{c} n_1, \ldots , n_k \\ \text {deg}(n_i)\le f(\epsilon _{i}) \end{array}} \frac{\chi _P(n_1 \cdots n_k)}{\prod _{i=1}^{k}|n_i|^{\frac{1}{2}+\varepsilon _i\alpha _i}}, \end{aligned} \end{aligned}$$

  (3.13)

  where $f(1)=g,$ and $f(-1)=g-1.$

• (3) Replace each product of $\varepsilon _f$-factors by its expected value when averaged over $\mathcal {P}_{2g+1,q}$.

  In our case, $\varepsilon _f$-factors are equal to 1. Thus, the product will not appear and will not affect the result.

• (4) Replace each summand by its expected value when averaged over $\mathcal {P}_{2g+1,q}$.

  We need first to average over all primes $P\in \mathcal {P}_{2g+1,q}.$ The next lemma gives the orthogonality relation for these quadratic Dirichlet characters over function fields.

Lemma 3.3

$$\begin{aligned} \lim _{\text {deg}(P) \rightarrow \infty } \frac{1}{\# \mathcal {P}_{2g+1,q}} \sum _{P\in \mathcal {P}_{2g+1,q}}\chi _P(n)= {\left\{ \begin{array}{ll} 1 &{}\quad \text{ if } n=\Box , \\ 0 &{}\quad \text{ otherwise. } \end{array}\right. } \end{aligned}$$

(3.14)

Proof

Consider the case when $n=\Box ,$ then we have

$$\begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} \chi _P(n)=\sum _{P\in \mathcal {P}_{2g+1,q}}\chi _P(l^2) = \sum _{\begin{array}{c} P\in \mathcal {P}_{2g+1,q}\\ P\not \mid l \end{array}} 1, \end{aligned}$$

(3.15)

since we are summing over primes of degree $2g+1$ and $P\not \mid l,$ and $\text {deg}(l)\le 2g,$ which means that we are counting all primes of degree $2g+1,$ thus,

$$\begin{aligned} \sum _{\begin{array}{c} P\in \mathcal {P}_{2g+1,q}\\ P\not \mid l \end{array}} 1 = \# \mathcal {P}_{2g+1,q}. \end{aligned}$$

(3.16)

Hence, if n is a square of a polynomial,

$$\begin{aligned} \lim _{\text {deg}(P)\rightarrow \infty } \frac{1}{\#\mathcal {P}_{2g+1,q}} \sum _{P\in \mathcal {P}_{2g+1,q}} \chi _P(n) = 1. \end{aligned}$$

(3.17)

It remains to consider the case when $n\ne \Box ,$ Rudnick [30] shows that

$$\begin{aligned} \left| \sum _{P\in \mathcal {P}_{2g+1,q}}\chi _P(n)\right| \ll \frac{|P|^{\frac{1}{2}}}{\log _q|P|} \text {deg}(n), \end{aligned}$$

(3.18)

and from Polynomial Prime Theorem (2.1), we have

$$\begin{aligned} \begin{aligned} \frac{1}{\#\mathcal {P}_{2g+1,q}} \sum _{P\in \mathcal {P}_{2g+1,q}} \chi _P(n)&\ll |P|^{-\frac{1}{2}} \text {deg}(n). \end{aligned} \end{aligned}$$

(3.19)

Hence, if n is not a square of a polynomial, we have that

$$\begin{aligned} \lim _{\text {deg}(P)\rightarrow \infty } \frac{1}{\#\mathcal {P}_{2g+1,q}} \sum _{P\in \mathcal {P}_{2g+1,q}} \chi _P(n) = 0. \end{aligned}$$

(3.20)

$\square$

Using Lemma 3.3, we can average the summand in (3.13) that is

$$\begin{aligned} \begin{aligned}&\lim _{\text {deg}(P) \rightarrow \infty } \frac{1}{\# \mathcal {P}_{2g+1,q}} \sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\begin{array}{c} n_1, \ldots , n_k \end{array}} \frac{\chi _P(n_1 \cdots n_k)}{\prod _{i=1}^{k}|n_i|^{\frac{1}{2}+\varepsilon _i\alpha _i}}\\&\quad = \sum _{m \text { monic}} \sum _{\begin{array}{c} n_1, \ldots , n_k \\ n_1\cdots n_k=m^2 \end{array}} \frac{1}{\prod _{i=1}^{k}|n_i|^{\frac{1}{2}+\varepsilon _i\alpha _i}}. \end{aligned} \end{aligned}$$

(3.21)

• (5) Let each $n_1, \ldots , n_k$ to be monic polynomials and call the total result $M_f(s, \alpha _1,\ldots , \alpha _k)$ to produce the desired conjecture.

  If we let

  $$\begin{aligned} R_k\left( \tfrac{1}{2};\varepsilon _1\alpha _1,\ldots , \varepsilon _k\alpha _k\right) = \sum _{m \text { monic}} \sum _{\begin{array}{c} n_1, \ldots , n_k \\ n_i \text { monic} \\ n_1\cdots n_k=m^2 \end{array}} \frac{1}{\prod _{i=1}^{k}|n_i|^{\frac{1}{2}+\varepsilon _i\alpha _i}}, \end{aligned}$$

  (3.22)

  then the extended sum produced by the recipe is

  $$\begin{aligned} M\left( \tfrac{1}{2};\alpha _1, \ldots , \alpha _k\right) = \sum _{\varepsilon _i=\pm 1} \prod _{i=1}^k \chi _P\left( \tfrac{1}{2}+\varepsilon _i \alpha _i\right) ^{-\frac{1}{2}} R_k\left( \tfrac{1}{2};\varepsilon _1\alpha _1,\ldots , \varepsilon _k\alpha _k\right) .\nonumber \\ \end{aligned}$$

  (3.23)

• (6) The conclusion is

  $$\begin{aligned} \begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} Z\left( \tfrac{1}{2};\alpha _1,\ldots , \alpha _k\right)&=\sum _{P\in \mathcal {P}_{2g+1,q}} M\left( \tfrac{1}{2},\alpha _1,\ldots , \alpha _k\right) \, \left( 1 + o(1)\right) .\\ \end{aligned} \end{aligned}$$

  (3.24)

3.3 Putting the conjecture in a more useful form

In this section, we put the conjecture (3.24) in a more useful form, we write $R_k$ as an Euler product, then factor out the appropriate $\zeta _A(s)$-factors. Let

$$\begin{aligned} \psi (x):= \sum _{\begin{array}{c} n_1, \ldots , n_k \\ n_i \text { monic} \\ n_1 \cdots n_k=x \end{array}} \frac{1}{|n_1|^{s+\alpha _1}\cdots |n_k|^{s+\alpha _k}}, \end{aligned}$$

(3.25)

then it is easy to see that $\psi (m^2)$ is multiplicative on m. We can write $R_k(s;\alpha _1,\ldots ,\alpha _k)$ as follows:

$$\begin{aligned} \begin{aligned} R_k(s;\alpha _1,\ldots ,\alpha _k)&= \sum _{m \text { monic}} \psi (m^2)\\&= \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \Bigg (1+ \sum _{j=1}^\infty \psi (P^{2j})\bigg ), \end{aligned} \end{aligned}$$

(3.26)

where

$$\begin{aligned} \psi (P^{2j}) = \sum _{\begin{array}{c} n_1, \ldots , n_k \\ n_i \text { monic} \\ n_1 \cdots n_k=P^{2j} \end{array}} \frac{1}{|n_1|^{s+\alpha _1}\cdots |n_k|^{s+\alpha _k}}. \end{aligned}$$

(3.27)

Since we have $n_1 \cdots n_k=P^{2j}$, then for each $i=1, \cdots , k,$ write $n_i$ as $n_i=P^{e_i},$ for some $e_i\ge 0$ and $e_1+\cdots +e_k=2j,$ and (3.27) becomes

$$\begin{aligned} \begin{aligned} \psi (P^{2j})&= \sum _{\begin{array}{c} e_1,\ldots ,e_k\ge 0 \\ e_1+\cdots +e_k=2j \end{array}} \prod _{i=1}^k \frac{1}{|P|^{e_i(s+\alpha _i)}}, \end{aligned} \end{aligned}$$

(3.28)

and so, we have

$$\begin{aligned} \begin{aligned} R_k(s;\alpha _1,\ldots ,\alpha _k)&= \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}}\Bigg (1+\sum _{j=1}^\infty \sum _{\begin{array}{c} e_1,\ldots ,e_k\ge 0 \\ e_1+\cdots +e_k=2j \end{array}} \prod _{i=1}^k \frac{1}{|P|^{e_i(s+\alpha _i)}}\Bigg ). \end{aligned} \end{aligned}$$

(3.29)

One can see that when $\alpha _i=0$ and $s=1/2$, the poles only arise from the terms with $e_1+\cdots +e_k=2$. Define $R_{k,P}(s;\alpha _1,\cdots ,\alpha _k)$ to be as follows:

$$\begin{aligned} \begin{aligned} R_{k,P}(s;\alpha _1,\ldots ,\alpha _k)&= 1+\sum _{j=1}^\infty \sum _{\begin{array}{c} e_1,\ldots ,e_k\ge 0 \\ e_1+\cdots +e_k=2j \end{array}} \prod _{i=1}^k \frac{1}{|P|^{e_i(s+\alpha _i)}}\\&= 1+\sum _{\begin{array}{c} e_1,\ldots ,e_k\ge 0 \\ e_1+\cdots +e_k=2 \end{array}} \prod _{i=1}^k \frac{1}{|P|^{e_i(s+\alpha _i)}}+ \text { (lower-order terms)}\\&= 1 + \sum _{1\le i\le j\le k} \frac{1}{|P|^{2s+\alpha _i+\alpha _j}} + O\Big ( |P|^{-4s+\epsilon }\Big ), \end{aligned} \end{aligned}$$

(3.30)

for $\mathfrak {R}(\alpha _i)$ small enough (see [7] for more details). And so, we have

$$\begin{aligned} \begin{aligned} R_{k,P}&(s;\alpha _1,\ldots ,\alpha _k)\\&= \prod _{1\le i\le j\le k} \left( 1+\frac{1}{|P|^{2s+\alpha _i+\alpha _j}}\right) \times \left( 1+O\left( |P|^{-4s+\epsilon }\right) \right) . \end{aligned} \end{aligned}$$

(3.31)

Recall that

$$\begin{aligned} \begin{aligned} \frac{\zeta _A(2s)}{\zeta _A(4s)}&= \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \Big (1+\frac{1}{|P|^{2s}}\Big )\\ \end{aligned} \end{aligned}$$

(3.32)

has a simple pole as $s=1/2.$ Therefore,

$$\begin{aligned} \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}}\Big (1+O\big (|P|^{-4s+\epsilon }\big )\Big ) \end{aligned}$$

(3.33)

is analytic in $\mathfrak {R}(s)>1/4,$ and $\prod _{P} R_{k,P}$ has a pole at $s=1/2$ of order $k(k+1)/2$ if $\alpha _i=0$ for all $i=1,\ldots ,k.$ It remains to factor out the appropriate zeta-factors. Since we have

$$\begin{aligned} R_k(s;\alpha _1,\ldots ,\alpha _k) =\prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} R_{k,P}(s;\alpha _1,\ldots ,\alpha _k), \end{aligned}$$

(3.34)

then from (3.31) and (3.32), we can write

$$\begin{aligned} \begin{aligned} R_k(s;\alpha _1,\ldots&,\alpha _k) = \prod _{1\le i\le j \le k} \zeta _A(2s+\alpha _i+\alpha _j) A_k(s;\alpha _1,\ldots ,\alpha _k), \end{aligned} \end{aligned}$$

(3.35)

where

$$\begin{aligned} \begin{aligned} A_k(s;\alpha _1&,\ldots ,\alpha _k) \\&= \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \left( R_{k,P}(s;\alpha _1,\ldots ,\alpha _k) \prod _{1\le i\le j \le k} \left( 1-\frac{1}{|P|^{2s+\alpha _i+\alpha _j}}\right) \right) .\quad \end{aligned} \end{aligned}$$

(3.36)

Notice that for some $\delta >0$ and for all $\alpha _i$’s in some sufficiently small neighbourhood of 0,  $A_k$ is an absolutely convergent Dirichlet series for $\mathfrak {R}(s)>1/2+\delta .$ Combining (3.23) and (3.35), we have

$$\begin{aligned} \begin{aligned} M\left( \tfrac{1}{2};\alpha _1, \ldots , \alpha _k\right) =&\sum _{\varepsilon _i=\pm 1} \prod _{i=1}^k \mathcal {X}_P\left( \tfrac{1}{2}+\varepsilon _i \alpha _i\right) ^{-\frac{1}{2}} \prod _{1\le i\le j \le k} \zeta _A(1+\alpha _i+\alpha _j) \\&\times A_k\left( \tfrac{1}{2};\varepsilon _1\alpha _1,\ldots ,\varepsilon _k\alpha _k\right) . \end{aligned} \end{aligned}$$

(3.37)

Hence,

$$\begin{aligned} \begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}}&Z\left( \tfrac{1}{2};\alpha _1, \ldots , \alpha _k\right) \\ =&\sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\varepsilon _i=\pm 1} \prod _{i=1}^k \mathcal {X}_P\left( \tfrac{1}{2}+\varepsilon _i \alpha _i\right) ^{-\frac{1}{2}} A_k\left( \tfrac{1}{2};\varepsilon _1\alpha _1,\ldots ,\varepsilon _k\alpha _k\right) \\&\times \prod _{1\le i\le j \le k} \zeta _A(1+\alpha _i+\alpha _j) \left( 1+o\left( 1\right) \right) . \end{aligned} \end{aligned}$$

(3.38)

From the definition of $\mathcal {X}_P(s)$ in (3.4), we have

$$\begin{aligned} \begin{aligned} \mathcal {X}_P\left( \tfrac{1}{2}+\varepsilon _i \alpha _i\right) ^{-\frac{1}{2}}&= |P|^{\frac{\varepsilon _i\alpha _i}{2}}\mathcal {X}\left( \tfrac{1}{2}+\varepsilon _i \alpha _i\right) ^{-\frac{1}{2}}.\\ \end{aligned} \end{aligned}$$

(3.39)

Hence,

$$\begin{aligned} \begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} Z\big (\tfrac{1}{2}&;\alpha _1, \cdots , \alpha _k\big ) \\ =&\sum _{\varepsilon _i=\pm 1} \prod _{i=1}^k \mathcal {X}\left( \tfrac{1}{2}+\varepsilon _i \alpha _i\right) ^{-\frac{1}{2}} \sum _{P\in \mathcal {P}_{2g+1,q}} R_k\left( \tfrac{1}{2};\varepsilon _1\alpha _1,\cdots ,\varepsilon _k\alpha _k\right) \\&\times |P|^{\frac{1}{2}\sum _{i=1}^k \varepsilon _i\alpha _i}\left( 1+o\big (1\big )\right) . \end{aligned} \end{aligned}$$

(3.40)

We finish this section writing $A_k$ as an Euler product in the following lemma.

Lemma 3.4

We have

$$\begin{aligned} \begin{aligned} A_k\left( \tfrac{1}{2};\alpha _1,\ldots ,\alpha _k\right)&= \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \prod _{1\le i\le j\le k}\left( 1-\frac{1}{|P|^{1+\alpha _i+\alpha _j}}\right) \\&\quad \times \frac{1}{2} \left( \prod _{i=1}^k \left( 1-\frac{1}{|P|^{1/2+\alpha _i}}\right) ^{-1}+ \prod _{i=1}^k \left( 1+\frac{1}{|P|^{1/2+\alpha _i}}\right) ^{-1}\right) . \\ \end{aligned} \end{aligned}$$

(3.41)

Proof

Applying $A_k(s;\alpha _1,\ldots ,\alpha _{k})$ and $R_{k,P}(s;\alpha _1,\ldots ,\alpha _{k})$ in (3.36) and (3.30) for $s=1/2$, we have

$$\begin{aligned} \begin{aligned}&A_k(\tfrac{1}{2};\alpha _1,\ldots ,\alpha _k)\\&\quad = \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \prod _{1\le i\le j \le k} \left( 1-\frac{1}{|P|^{1+\alpha _i+\alpha _j}}\right) \Bigg (1+\sum _{j=1}^\infty \sum _{\begin{array}{c} e_1,\ldots ,e_k\ge 0 \\ e_1+\cdots +e_k=2j \end{array}} \prod _{i=1}^k \frac{1}{|P|^{e_i(1/2+\alpha _i)}}\Bigg ). \end{aligned} \end{aligned}$$

(3.42)

By simplifying the second brackets, we obtain the result in the lemma, i.e.

$$\begin{aligned} \begin{aligned}&1+ \sum _{j=1}^\infty \sum _{\begin{array}{c} e_1,\cdots ,e_k\ge 0 \\ e_1+\cdots +e_k=2j \end{array}} \prod _{i=1}^k \Bigg (\frac{1}{|P|^{(1/2+\alpha _i)}}\Bigg )^{e_i} \\&\quad = \sum _{j=0}^\infty \frac{1}{2} \Bigg (2 \sum _{\begin{array}{c} e_1,\cdots ,e_k\ge 0 \\ e_1+\cdots +e_k=2j \end{array}} \prod _{i=1}^k \Bigg (\frac{1}{|P|^{(1/2+\alpha _i)}}\Bigg )^{e_i}\Bigg )\\&\quad = \frac{1}{2} \Bigg ( \prod _{i=1}^k \sum _{e_i= 0 }^\infty \Bigg (\frac{1}{|P|^{(1/2+\alpha _i)}}\Bigg )^{e_i} + \prod _{i=1}^k \sum _{e_i= 0 }^\infty (-1)^{e_i} \Bigg (\frac{1}{|P|^{(1/2+\alpha _i)}}\Bigg )^{e_i}\Bigg )\\&\quad = \frac{1}{2} \Bigg (\prod _{i=1}^k \Big (1-\frac{1}{|P|^{1/2+\alpha _i}}\Big )^{-1}+ \prod _{i=1}^k \Big (1+\frac{1}{|P|^{1/2+\alpha _i}}\Big )^{-1}\Bigg ). \end{aligned} \end{aligned}$$

(3.43)

$\square$

3.4 The contour integral representation of the conjecture

We begin this section with Lemma 2.5.2 from [7], which helps write our conjecture as a contour integral.

Lemma 3.5

Suppose F is a symmetric function of k variables, regular near $(0,\ldots ,0)$, and that f(s) has a simple pole $s=0$ of residue 1 and is otherwise analytic in a neighbourhood of $s=0,$ and let

$$\begin{aligned} K(a_1,\ldots ,a_k)=F(a_1,\ldots ,a_k) \prod _{1\le i\le j\le k} f(a_i+a_j), \end{aligned}$$

(3.44)

or

$$\begin{aligned} K(a_1,\ldots ,a_k)=F(a_1,\ldots ,a_k) \prod _{1\le i< j\le k} f(a_i+a_j). \end{aligned}$$

(3.45)

If $\alpha _i+\alpha _j$ are contained in the region of analyticity of f(s),  then

$$\begin{aligned} \begin{aligned} \sum _{\varepsilon _i=\pm 1} K(\varepsilon _1a_1,\ldots ,\varepsilon _k a_k)&= \frac{(-1)^{k(k-1)/2}}{(2\pi i)^k} \frac{2^k}{k!}\oint \cdots \oint K(z_1,\ldots ,z_k)\\&\quad \times \frac{\Delta (z_1^2,\ldots ,z_k^2)^2 \prod _{i=1}^kz_i}{\prod _{i=1}^k \prod _{j=1}^k (z_i-\alpha _j)(z_i+\alpha _j)}\mathrm{{d}}z_1\cdots \mathrm{{d}}z_k, \end{aligned} \end{aligned}$$

(3.46)

and

$$\begin{aligned} \begin{aligned}&\sum _{\varepsilon _i=\pm 1} \Bigg (\prod _{i=1}^k \varepsilon _i\Bigg ) K(\varepsilon _1a_1,\ldots , \varepsilon _k a_k)\\&\quad = \frac{(-1)^{k(k-1)/2}}{(2\pi i)^k} \frac{2^k}{k!}\oint \cdots \oint K(z_1,\ldots ,z_k)\\&\quad \quad \times \frac{\Delta (z_1^2,\cdots ,z_k^2)^2 \prod _{i=1}^k \alpha _i}{\prod _{i=1}^k \prod _{j=1}^k (z_i-\alpha _j)(z_i+\alpha _j)}\mathrm{{d}}z_1\cdots \mathrm{{d}}z_k, \end{aligned} \end{aligned}$$

(3.47)

where the path of the integration encloses the $\pm \alpha _i$s.

Recall that

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} Z\left( \tfrac{1}{2};\alpha _1,\ldots ,\alpha _k\right) \\&\quad =\sum _{P\in \mathcal {P}_{2g+1,q}}\prod _{i=1}^k \mathcal {X}_P\left( \tfrac{1}{2}+\alpha _i\right) ^{-\frac{1}{2}} L\left( \tfrac{1}{2}+\alpha _i,\chi _P\right) , \end{aligned} \end{aligned}$$

(3.48)

where $\mathcal {X}_P(s)$ is defined in (3.4). Since $\mathcal {X}_P(\tfrac{1}{2}+\alpha _i)^{-\frac{1}{2}}$ does not depend on P, we can factor out it, and from (3.48) and (3.40), we have

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \prod _{i=1}^k L\left( \tfrac{1}{2}+\alpha _i,\chi _P\right) \\&\quad = \sum _{P\in \mathcal {P}_{2g+1,q}} |P|^{-\frac{1}{2}\sum _{i=0}^k \alpha _i}\prod _{i=1}^k \mathcal {X}\left( \tfrac{1}{2}+\alpha _i\right) ^{\frac{1}{2}} \sum _{\varepsilon _i=\pm 1} \prod _{i=1}^k \mathcal {X}\left( \tfrac{1}{2}+\varepsilon _i\alpha _i\right) ^{-\frac{1}{2}}\\&\quad \quad \times A_k\left( \tfrac{1}{2};\alpha _1,\cdots ,\alpha _k\right) |P|^{\frac{1}{2}\sum _{i=0}^k \varepsilon _i \alpha _i} \\&\quad \quad \times \prod _{1\le i< j\le k}\zeta _A(1+\varepsilon _i\alpha _i+\varepsilon _j\alpha _j)\Big (1+o\big (1\big )\Big ). \end{aligned} \end{aligned}$$

(3.49)

From each term in the second product, we factor out $(\log q)^{-1}$ to get

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \prod _{i=1}^k L\left( \tfrac{1}{2}+\alpha _i,\chi _P\right) \\&\quad = \sum _{P\in \mathcal {P}_{2g+1,q}} \frac{|P|^{-\frac{1}{2}\sum _{i=0}^k \alpha _i}\prod _{i=1}^k \mathcal {X}\left( \tfrac{1}{2}+\alpha _i\right) ^{\frac{1}{2}}}{(\log q)^{k(k+1)/2}} \sum _{\varepsilon _i=\pm 1} \prod _{i=1}^k \mathcal {X}\left( \tfrac{1}{2}+\varepsilon _i\alpha _i\right) ^{-\frac{1}{2}}\\&\quad \quad \times A_k\left( \tfrac{1}{2};\alpha _1,\ldots ,\alpha _k\right) |P|^{\frac{1}{2}\sum _{i=0}^k \varepsilon _i \alpha _i} \\&\quad \quad \times \prod _{1\le i< j\le k}\zeta _A(1+\varepsilon _i\alpha _i+\varepsilon _j\alpha _j)(\log q) \Big (1+o\big (1\big )\Big ). \end{aligned} \end{aligned}$$

(3.50)

Now, call

$$\begin{aligned} F(\alpha _1,\ldots ,\alpha _k)= \prod _{i=1}^k\mathcal {X}(\tfrac{1}{2}+\alpha _i)^{-\frac{1}{2}} A_k(\tfrac{1}{2};\alpha _i,\ldots ,\alpha _k) |P|^{\frac{1}{2}\sum _{i=1}^k\alpha _i} \end{aligned}$$

(3.51)

and

$$\begin{aligned} f(s)=\zeta _A(1+s)\log q \; \text { and so }\; f(\alpha _i+\alpha _j)=\zeta _A(1+\alpha _i+\alpha _j)\log q, \end{aligned}$$

(3.52)

where f(s) has a simple pole at $s=0$ with residue 1.

If we denote

$$\begin{aligned} K(\alpha _1,\ldots ,\alpha _k)= F(\alpha _1,\ldots ,\alpha _k) \prod _{1\le i\le j\le k}f(\alpha _i+\alpha _j), \end{aligned}$$

(3.53)

then (3.50) can be written as follows:

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \prod _{i=1}^k L \left( \tfrac{1}{2}+\alpha _i,\chi _P\right) \\&\quad =\sum _{P\in \mathcal {P}_{2g+1,q}} \frac{\prod _{i=1}^k|P|^{-\frac{1}{2}\sum _{i=0}^k \alpha _i} \mathcal {X}\left( \tfrac{1}{2}+\alpha _i \right) ^{\frac{1}{2}}}{(\log q)^{k(k+1)/2}} \\&\quad \quad \times \sum _{\varepsilon _i=\pm 1} K(\varepsilon _1\alpha _1,\ldots ,\varepsilon _k\alpha _k) \left( 1+o\left( 1\right) \right) .\\ \end{aligned} \end{aligned}$$

(3.54)

Using Lemma 3.5, we have

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \prod _{i=1}^k L\left( \tfrac{1}{2}+\alpha _i,\chi _P\right) \\&\quad =\sum _{P\in \mathcal {P}_{2g+1,q}} \frac{\prod _{i=1}^k|P|^{-\frac{1}{2}\sum _{i=0}^k \alpha _i} \mathcal {X}\left( \tfrac{1}{2}+\alpha _i\right) ^{\frac{1}{2}}}{(\log q)^{k(k+1)/2}} \frac{(-1)^{k(k-1)/2}}{(2\pi i)^k} \frac{2^k}{k!}\\&\quad \quad \times \oint \cdots \oint K(z_1,\ldots ,z_k) \frac{\Delta (z_1^2,\ldots ,z_k^2)^2 \prod _{i=1}^kz_i}{\prod _{i=1}^k \prod _{j=1}^k (z_i-\alpha _j)(z_i+\alpha _j)}\mathrm{{d}}z_1\cdots \mathrm{{d}}z_k\\&\quad \quad \times \Big (1+o\big (1\big )\Big )\\&\quad =\sum _{P\in \mathcal {P}_{2g+1,q}} \prod _{i=1}^k |P|^{-\frac{1}{2}\sum _{i=0}^k \alpha _i} \mathcal {X}\left( \tfrac{1}{2}+\alpha _i\right) ^{\frac{1}{2}} \frac{(-1)^{k(k-1)/2}}{(2\pi i)^k} \frac{2^k}{k!}\\&\quad \quad \times \oint \cdots \oint F(z_1,\ldots ,z_k) \prod _{1\le i \le j\le k}\zeta _A(1+\varepsilon _i\alpha _i+\varepsilon _j\alpha _j)\\&\quad \quad \times \frac{\Delta (z_1^2,\ldots ,z_k^2)^2 \prod _{i=1}^kz_i}{\prod _{i=1}^k \prod _{j=1}^k (z_i-\alpha _j)(z_i+\alpha _j)}\mathrm{{d}}z_1\cdots \mathrm{{d}}z_k +o\big (|P|\big )\\&\quad =\sum _{P\in \mathcal {P}_{2g+1,q}} \prod _{i=1}^k |P|^{-\frac{1}{2}\sum _{i=0}^k \alpha _i} \mathcal {X}\left( \tfrac{1}{2}+\alpha _i\right) ^{\frac{1}{2}} \frac{(-1)^{k(k-1)/2}}{(2\pi i)^k} \frac{2^k}{k!}\\&\quad \quad \times \oint \cdots \oint K(z_1,\ldots ,z_k) \frac{\Delta (z_1^2,\ldots ,z_k^2)^2 \prod _{i=1}^kz_i}{\prod _{i=1}^k \prod _{j=1}^k (z_i-\alpha _j)(z_i+\alpha _j)}\mathrm{{d}}z_1\cdots dz_k \\&\quad \quad +o\big (|P|\big ) \end{aligned} \end{aligned}$$

(3.55)

with

$$\begin{aligned} K(z_1,\ldots ,z_k) = F(z_1,\ldots ,z_k) \prod _{1\le i \le j\le k}\zeta _A(1+\varepsilon _i\alpha _i+\varepsilon _j\alpha _j). \end{aligned}$$

(3.56)

Moreover, if we denote

$$\begin{aligned} \begin{aligned} G(z_1,\ldots ,z_k)&= \prod _{i=1}^k\mathcal {X}\left( \tfrac{1}{2}+\alpha _i\right) ^{-\frac{1}{2}} A_k\left( \tfrac{1}{2};\alpha _i,\ldots ,\alpha _k\right) \prod _{1\le i \le j\le k}\zeta _A(1+z_i+z_j) \end{aligned} \end{aligned}$$

(3.57)

then (3.55) becomes

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \prod _{i=1}^k |P|^{-\frac{1}{2}\sum _{i=0}^k \alpha _i} \mathcal {X}\left( \tfrac{1}{2}+\alpha _i\right) ^{\frac{1}{2}} \frac{(-1)^{k(k-1)/2}}{(2\pi i)^k} \frac{2^k}{k!} \\&\quad \times \oint \cdots \oint G(z_1,\ldots ,z_k) |P|^{\frac{1}{2}\sum _{i=0}^k z_i}\frac{\Delta (z_1^2,\ldots ,z_k^2)^2 \prod _{i=1}^kz_i}{\prod _{i=1}^k \prod _{j=1}^k (z_i-\alpha _j)(z_i+\alpha _j)}\mathrm{{d}}z_1\cdots \mathrm{{d}}z_k\\&\quad +\,o\big (|P|\big ). \end{aligned} \end{aligned}$$

(3.58)

Now, letting $\alpha _i\rightarrow 0,$ we have

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} L(\tfrac{1}{2},\chi _P)^k\\&\quad =\sum _{P\in \mathcal {P}_{2g+1,q}} \frac{(-1)^{k(k-1)/2}}{(2\pi i)^k} \frac{2^k}{k!} \oint \cdots \oint G(z_1,\ldots ,z_k) |P|^{\frac{1}{2}\sum _{i=0}^k z_i} \\&\quad \quad \times \frac{\Delta (z_1^2,\ldots ,z_k^2)^2 \prod _{i=1}^kz_i}{\prod _{i=1}^k z_i^{2k}}\mathrm{{d}}z_1\cdots \mathrm{{d}}z_k +o\big (|P|\big ). \end{aligned} \end{aligned}$$

(3.59)

Calling

$$\begin{aligned} \begin{aligned} Q_k(x)&= \frac{(-1)^{k(k-1)/2}}{(2\pi i)^k} \frac{2^k}{k!} \oint \cdots \oint G(z_1,\ldots ,z_k) \\&\quad \times q^{\frac{x}{2}\sum _{i=0}^k z_i}\frac{\Delta (z_1^2,\ldots ,z_k^2)^2 \prod _{i=1}^kz_i}{\prod _{i=1}^k z_i^{2k}}\mathrm{{d}}z_1\cdots \mathrm{{d}}z_k, \end{aligned} \end{aligned}$$

(3.60)

we obtain the formula of the Conjecture 2.2, i.e.

$$\begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} L(\tfrac{1}{2},\chi _P)^k= \sum _{P\in \mathcal {P}_{2g+1,q}} Q_k(\log _q|P|)\left( 1+o\left( 1\right) \right) . \end{aligned}$$

(3.61)

4 Some conjectural formulae for moments of L-functions associated with $\chi _P$

We use Conjecture 2.2 to obtain explicit conjectural values for several moments of quadratic Dirichlet L-functions associated with $\chi _P$ over function fields.

4.1 First moment

We will use Conjecture 2.2 when $k=1$ to compute the first moment of our family of L-functions, then compare the result with that of Andrade and Keating proved in [3]. For $k=1$, the formula in Conjecture 2.2 gives

$$\begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _P\right) = \sum _{P\in \mathcal {P}_{2g+1,q}} Q_1\big (\log _q|P|\big )\big (1+o(1)\big ), \end{aligned}$$

(4.1)

where $Q_1(x)$ is polynomial of degree 1. From the contour integral formula for $Q_k(x)$ in (2.14), we have

$$\begin{aligned} Q_1(x)= \frac{1}{\pi i} \oint \frac{G(z_1)\Delta (z_1^2)^2}{z_1} \, q^{\frac{x}{2}z_1} \, \mathrm{{d}}z_1, \end{aligned}$$

(4.2)

where

$$\begin{aligned} G(z_1) = A\left( \tfrac{1}{2};z_1\right) \mathcal {X}\left( \tfrac{1}{2}+z_1\right) ^{-\frac{1}{2}}\zeta _A(1+2z_1). \end{aligned}$$

(4.3)

Recall that, the Vandermonde determinant is defined

$$\begin{aligned} \Delta (z_1,\ldots ,z_k) = \prod _{1\le i< j\le k}(z_j-z_i), \end{aligned}$$

(4.4)

which for $k=1$ is equal to

$$\begin{aligned} \Delta (z_1^2)^2=1, \end{aligned}$$

(4.5)

and

$$\begin{aligned} \mathcal {X}\left( \tfrac{1}{2}+z_1\right) ^{-\frac{1}{2}}=q^{-z_1/2}. \end{aligned}$$

(4.6)

Therefore, (4.2) becomes

$$\begin{aligned} \begin{aligned} Q_1(x)=\frac{1}{\pi i} \oint \frac{A\left( \tfrac{1}{2};z_1\right) \zeta _A(1+2z_1)}{z_1} \, q^{\frac{x-1}{2}z_1} \, \mathrm{{d}}z_1, \end{aligned} \end{aligned}$$

(4.7)

with

$$\begin{aligned} \begin{aligned} A\left( \tfrac{1}{2};z_1\right)&= \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \left( 1-\frac{1}{|P|^{1+2z_1}}\right) \\&\quad \times \frac{1}{2} \left( \left( 1-\frac{1}{|P|^{1/2+z_1}}\right) ^{-1}+\left( 1+\frac{1}{|P|^{1/2+z_1}}\right) ^{-1}\right) . \end{aligned} \end{aligned}$$

(4.8)

In order to compute the integral in (4.7) where the contour is a small circle around the origin, we need to locate the poles of the integrand. So let

$$\begin{aligned} f(z_1)=\frac{A\left( \tfrac{1}{2};z_1\right) \zeta _A(1+2z_1)}{z_1} \, q^{\frac{x-1}{2}z_1}, \end{aligned}$$

(4.9)

note that the zeta-function $\zeta _A(1+2z_1)$ has a simple pole at $z_1=0,$ which means that $f(z_1)$ has a pole of order 2 at $z_1=0.$ We compute the residue by expand $f(z_1)$ as a Laurent series and consider the coefficient of $z_1^{-1}.$ Expanding the numerator of $f(z_1)$ around $z_1=0$, we have

• (1)
  $$A\left( \tfrac{1}{2};z_1\right) = A\left( \tfrac{1}{2};0\right) + A'\left( \tfrac{1}{2};0\right) z_1+ \frac{1}{2}A''\left( \tfrac{1}{2};0\right) z_1^2+\cdots $$
• (2)
  $$\zeta _A(1+2z_1) = \frac{1}{2\log q}\frac{1}{z_1} +\frac{1}{2}+\frac{1}{6} (\log q) z_1 -\frac{1}{90} (\log q)^3 z_1^3 + \cdots $$
• (3)
  $$q^{\frac{x-1}{2}z_1}= 1+\frac{1}{2} (x-1) (\log q) z_1 +\frac{1}{8} (x-1)^2 (\log q)^2 z_1^2+ \cdots $$

Hence, $f(z_1)$ can be written as follows:

$$\begin{aligned} \begin{aligned} f(z_1)&= \Big ( A(\tfrac{1}{2};0)\frac{1}{z_1} + A'(\tfrac{1}{2};0)+ \frac{1}{2}A''(\tfrac{1}{2};0)z_1+\cdots \Big )\\&\quad \times \Big (\frac{1}{2\log q}\frac{1}{z_1} +\frac{1}{2}+\frac{1}{6} (\log q) z_1 -\frac{1}{90} (\log q)^3 z_1^3 + \cdots \Big )\\&\quad \times \Big ( 1 -\frac{1}{2} (\log q) z_1+ \frac{1}{8} (\log q)^2 z_1^2+\cdots \Big )\\&\quad \times \Big (1 +\frac{1}{2} (\log q)x z_1+ \frac{1}{8} (\log q)^2 x^2 z_1^2+\cdots \Big ).\\ \end{aligned} \end{aligned}$$

(4.10)

Considering the coefficient of $z_1^{-1}$, we have

$$\begin{aligned} \underset{z_1=0}{\text {Res}} f(z_1)= \frac{1}{4}(1+x) A\left( \tfrac{1}{2};0\right) + \frac{1}{2\log q} A'\left( \tfrac{1}{2};z_1\right) . \end{aligned}$$

(4.11)

After straightforward calculations, using the definition for $A_k\left( \tfrac{1}{2},z_1,\ldots ,z_k\right)$, we have

$$\begin{aligned} \begin{aligned} A\left( \tfrac{1}{2};z_1\right) = 1 \ \ \text {and} \ \ \ A'\left( \tfrac{1}{2};z_1\right) = 0, \end{aligned} \end{aligned}$$

(4.12)

and so

$$\begin{aligned} \underset{z_1=0}{\text {Res}} f(z_1)=\frac{1}{4}(1+x). \end{aligned}$$

(4.13)

Hence, we have

$$\begin{aligned} \begin{aligned} Q_1(x)&= \frac{1}{4\pi i}(1+x) \oint 1 \, \mathrm{{d}}z_1\\&= \frac{1}{2} (1+x). \end{aligned} \end{aligned}$$

(4.14)

Finally, we can write the first moment as follows:

$$\begin{aligned} \begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _P\right)&= \sum _{P\in \mathcal {P}_{2g+1,q}} Q_1(\log _q|P|) \left( 1+o\left( 1\right) \right) \\&= \sum _{P\in \mathcal {P}_{2g+1,q}} \frac{1}{2}\left( 1+\log _q|P|\right) \left( 1+o\left( 1\right) \right) \\&= \frac{|P|}{2\log _q|P|} \left( 1+ \log _q|P|\right) +o\big (|P|\big ). \end{aligned} \end{aligned}$$

(4.15)

If we compare Theorem 2.4 of [3] with the conjecture, we can see that the main term and the principal lower-order terms are the same. In other words, Theorem 2.4 of [3] proves our conjecture with an error $O\big (|P|^{3/4+\epsilon }\big ).$ In the next two sections, we use our conjecture to determine the asymptotic of the second and third moments of our family of L-functions, and it can be seen that the polynomials $Q_2(x)$ and $Q_3(x)$ in (4.22) and (4.25) are similar to ones in [4].

4.2 Second moment

For $k=2$, the conjecture 2.2 gives

$$\begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _P\right) ^2= \sum _{P\in \mathcal {P}_{2g+1,q}} Q_2\left( \log _q|P|\right) \left( 1+o(1)\right) , \end{aligned}$$

(4.16)

where $Q_2(x)$ is a polynomial of degree 3, given by

$$\begin{aligned} Q_2(x)= \frac{-1}{2\pi ^2} \oint \oint \frac{G(z_1,z_2)\Delta (z_1^2,z_2^2)^2}{z_1^3z_2^3} \, q^{\frac{x}{2}(z_1+z_2)} \, \mathrm{{d}}z_1 \,\mathrm{{d}}z_2, \end{aligned}$$

(4.17)

with

$$\begin{aligned} G(z_1,z_2)&= A\left( \tfrac{1}{2};z_1,z_2\right) \mathcal {X}\left( \tfrac{1}{2}+z_1\right) ^{-\frac{1}{2}} \mathcal {X}\left( \tfrac{1}{2}+z_2\right) ^{-\frac{1}{2}} \nonumber \\&\quad \times \, \zeta _A(1+2z_1)\zeta _A(1+z_1+z_2)\zeta _A(1+2z_2), \end{aligned}$$

(4.18)

$$\begin{aligned}&\mathcal {X}\left( \tfrac{1}{2}+z_1\right) ^{-\frac{1}{2}}\mathcal {X}\left( \tfrac{1}{2}+z_2\right) ^{-\frac{1}{2}} =q^{-\frac{1}{2}(z_1+z_2)}, \end{aligned}$$

(4.19)

and

$$\begin{aligned} \begin{aligned} \Delta \left( z_1^2,z_2^2\right) ^2 =\left( z_2^2-z_1^2\right) ^2. \end{aligned} \end{aligned}$$

(4.20)

If

$$\begin{aligned} \begin{aligned} f(z_1,z_2)&=\frac{A\left( \tfrac{1}{2};z_1,z_2\right) \zeta _A(1+2z_1)\zeta _A(1+z_1+z_2)\zeta _A(1+2z_2)(z_2^2-z_1^2)^2}{z_1^3z_2^3}\\&\quad \times q^{\frac{x-1}{2}(z_1+z_2)}, \end{aligned} \end{aligned}$$

(4.21)

then we have

$$\begin{aligned} \begin{aligned} Q_2(x)&=\frac{-1}{2\pi ^2} \oint \oint f(z_1,z_2) \, \mathrm{{d}}z_1\,\mathrm{{d}}z_2\\&= \frac{1}{24 \log ^3(q)} \Big ((x^3+6x^2+11x+6) A(1/2;0,0) \log ^3(q) \\&\quad +\, (3x^2+12x+11)\\&\quad \times \log ^2(q) (A_1(\tfrac{1}{2};0,0)+A_2(\tfrac{1}{2};0,0)) +12(2+x) \log (q) A_{12}(\tfrac{1}{2};0,0) \\&\quad -\, 2 (A_{222}(\tfrac{1}{2};0,0) -3A_{122}(\tfrac{1}{2};0,0) -3A_{112}(\tfrac{1}{2};0,0) +A_{111}(\tfrac{1}{2};0,0))\Big ), \end{aligned} \end{aligned}$$

(4.22)

where $A_j$ is the partial derivative, evaluates at zero, of the function $A\left( \tfrac{1}{2};z_1,\cdots ,z_k\right)$ with respect to jth variable, with indices denoting higher derivatives, i.e.

$$\begin{aligned} A_{122}\left( \tfrac{1}{2};0,\ldots ,0\right) =\frac{\partial }{\partial z_1}\frac{\partial ^2}{\partial z_2^2}A\left( \tfrac{1}{2};z_1,\ldots ,z_k\right) \Bigg |_{z_1=z_2=\cdots =z_k=0}. \end{aligned}$$

Hence, we can write the leading order asymptotic for the second moment for the family of L-function when $g\rightarrow \infty$ as

$$\begin{aligned} \begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}}L(\tfrac{1}{2},\chi _P)^2&\sim \frac{1}{24 \zeta _A(2)} |P| (\log _q|P|)^2. \end{aligned} \end{aligned}$$

(4.23)

Comparing with Andrade and Keating result (Theorem 2.5 of [3]), we see that their theorem proves our conjecture with an error $O\left( |P| \log _q|P|\right) .$

4.3 Third moment

For the third moment, Conjecture 2.2 states that

$$\begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _P\right) ^3= \sum _{P\in \mathcal {P}_{2g+1,q}} Q_3\big (\log _q|P|\big )\big (1+o(1)\big ), \end{aligned}$$

(4.24)

where $Q_3(x)$ is a polynomial of degree 3.

Thus, with the help of the symbolic manipulation software Mathematica, we compute the triple contour integral and obtain

$$\begin{aligned} Q_3(x)&=\frac{1}{8640 \log ^6(q)}\Bigg (3 (x+3)^2 \left( x^4+12 x^3+49 x^2+78 x+40\right) A(0,0,0) \log ^6(q)\\&\quad +\,\,4 \left( 3 x^5+45 x^4+260 x^3+720 x^2+949 x+471\right) \Big (A_{3}(0,0,0)+A_{2}(0,0,0)\\&\quad +A_{1}(0,0,0)\Big ) \log ^5(q)+4 \left( 15 x^4+180 x^3 +780 x^2+1440 x+949\right) \Big (A_{23}(0,0,0)\\&\quad +A_{13}(0,0,0)+A_{12}(0,0,0)\Big ) \log ^4(q)-\,10 \left( x^3+9 x^2+26 x+24\right) \Big (2 A_{333}(0,0,0)\\&\quad -\,3 A_{233}(0,0,0)-3 A_{223}(0,0,0)\\&\quad +\,2 A_{222}(0,0,0)-3 A_{133}(0,0,0)-36 A_{123}(0,0,0)\\&\quad -\,3 A_{122}(0,0,0)-3 A_{113}(0,0,0)-3 A_{112}(0,0,0)+2 A_{111}(0,0,0)\Big ) \log ^3(q)\\&\quad -\,20 \left( 3 x^2+18 x+26\right) \Big (A_{2333}(0,0,0)+A_{2223}(0,0,0)\\&\quad +\,A_{1333}(0,0,0)-6 A_{1233}(0,0,0)\\&\quad -\,6 A_{1223}(0,0,0)+A_{1222}(0,0,0)-6 A_{1123}(0,0,0)\\&\quad +\,\,A_{1113}(0,0,0)+A_{1112}(0,0,0)\Big ) \\&\quad \times \log ^2(q)+6 (x+3) \Big (2 A_{33333}(0,0,0)-5 A_{23333}(0,0,0)\\&\quad -\,10 A_{22333}(0,0,0)-10 A_{22233}(0,0,0)\\&\quad -\,5 A_{22223}(0,0,0)+2 A_{22222}(0,0,0)-5 A_{13333}(0,0,0)\\&\quad +\,60 A_{12233}(0,0,0)-5 A_{12222}(0,0,0)\\&\quad -\,10 A_{11333}(0,0,0)+60 A_{11233}(0,0,0)\\&\quad +\,60 A_{11223}(0,0,0)-10 A_{11222}(0,0,0)\\&\quad -\,10 A_{11133}(0,0,0)-10 A_{11122}(0,0,0)\\&\quad -\,5 A_{11113}(0,0,0)-5 A_{11112}(0,0,0) +2 A_{11111}(0,0,0)\Big ) \log (q)\\ \end{aligned}$$

$$\begin{aligned}&\quad \quad +4 \Big (3 A_{233333}(0,0,0)-20 A_{222333}(0,0,0)+3 A_{222223}(0,0,0)\nonumber \\&\quad \quad +3 A_{133333}(0,0,0)-30 A_{123333}(0,0,0)\nonumber \\&\quad \quad +30 A_{122333}(0,0,0)+30 A_{122233}(0,0,0)\nonumber \\&\quad \quad -30 A_{122223}(0,0,0)+3 A_{122222}(0,0,0)\nonumber \\&\quad \quad +30 A_{112333}(0,0,0)+30 A_{112223}(0,0,0)\nonumber \\&\quad \quad -20 A_{111333}(0,0,0)+30 A_{111233}(0,0,0)\nonumber \\&\quad \quad +30 A_{111223}(0,0,0)-20 A_{111222}(0,0,0)\nonumber \\&\quad \quad -30 A_{111123}(0,0,0)+3 A_{111113}(0,0,0)+3 A_{111112}(0,0,0)\Big )\Bigg ), \end{aligned}$$

(4.25)

where $A\left( \tfrac{1}{2};z_1,z_2,z_3\right)$ is defined in Lemma 3.4. Hence, the leading order asymptotic for the third moment for our family of L-functions is given by

$$\begin{aligned} \begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}}L\left( \tfrac{1}{2},\chi _P\right) ^3&\sim \frac{1}{2880} |P| A(\tfrac{1}{2};0,0,0) (\log _q|P|)^5, \end{aligned} \end{aligned}$$

(4.26)

where

$$\begin{aligned} \begin{aligned} A(\tfrac{1}{2};0,0,0) = \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \left( 1- \frac{6 |P|^2-8|P|+3}{|P|^4}\right) .\\ \end{aligned} \end{aligned}$$

(4.27)

4.4 Leading order for general k

The main aim in this section is to obtain a conjecture for the leading order asymptotics of the moments for a general integer k. The calculations presented here are based on the calculations first presented in [1, 26]. To obtain the main formula we need the following lemma.

Lemma 4.1

Let F be a symmetric function of k variables, regular near $(0,\cdots ,0)$ and f(s) has a simple pole of residue 1 at $s=0$ and analytic in a neighbourhood of $s=0.$ Let

$$\begin{aligned} \begin{aligned} K\left( |P|;w_1,\ldots ,w_k\right)&= \sum _{\varepsilon _i=\pm 1}e^{\frac{1}{2} \log |P| \sum _{i=1}^k \varepsilon _i w_i} F\left( \varepsilon _1 w_1,\ldots ,\varepsilon _k w_k\right) \\&\quad \times \prod _{1 \le i \le j \le k} f\left( \varepsilon _i w_i + \varepsilon _j w_j\right) , \end{aligned} \end{aligned}$$

(4.28)

and define $I\left( |P|,k;w=0\right)$ to be the value of K when $w_1,\ldots ,w_k=0.$ We have that

$$\begin{aligned} I\left( |P|,k;0\right) \sim \left( \frac{1}{2}\log |P|\right) ^{k(k+1)/2}F(0,\ldots ,0) 2^{k(k+1)/2}. \left( \prod _{i=1}^k \frac{i!}{\left( 2i\right) !}\right) . \end{aligned}$$

(4.29)

Proof

See Lemma 5 in [4].

We are in a position to obtain the desired formula, from (3.55) recall that

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \prod _{i=1}^k L(\tfrac{1}{2}+\alpha _i,\chi _P)\\&\quad =\sum _{P\in \mathcal {P}_{2g+1,q}} \prod _{i=1}^k\frac{|P|^{-\frac{1}{2}\sum _{i=0}^k \alpha _i} \mathcal {X}(\tfrac{1}{2}+\alpha _i)^{\frac{1}{2}}}{(\log q)^{k(k+1)/2}} \sum _{\varepsilon _i=\pm 1} K(\varepsilon _1\alpha _1,\ldots ,\varepsilon _k\alpha _k)\Big (1+o\big (1\big )\Big ),\\ \end{aligned} \end{aligned}$$

(4.30)

where

$$\begin{aligned} \begin{aligned}&K(\varepsilon _1\alpha _1,\ldots ,\varepsilon _k\alpha _k)\\&\quad =\sum _{\varepsilon _i=\pm 1} \prod _{i=1}^k \mathcal {X}(\tfrac{1}{2}+\varepsilon _i\alpha _i)^{-\frac{1}{2}} A_k(\tfrac{1}{2};\alpha _1,\ldots ,\alpha _k) |P|^{\frac{1}{2}\sum _{i=0}^k \varepsilon _i \alpha _i} \\&\quad \quad \times \prod _{1\le i< j\le k}\zeta _A(1+\varepsilon _i\alpha _i+\varepsilon _j\alpha _j)(\log q). \end{aligned} \end{aligned}$$

(4.31)

Applying the above Lemma with

$$\begin{aligned} \begin{aligned} f(s)&= \zeta _A(1+s) \log q,\\ F\left( w_1,\ldots , w_k\right)&= \prod _{i=1}^k \mathcal {X}(\tfrac{1}{2}+\alpha _i)^{-\frac{1}{2}} A_k\left( \tfrac{1}{2};w_1,\ldots , w_k\right) ,\\ K\left( |P|;w_1,\ldots ,w_k\right)&= \sum _{\varepsilon _i=\pm 1}|P|^{\frac{1}{2} \sum _{i=1}^k \varepsilon _i w_i} F\left( \varepsilon _1 w_1,\ldots ,\varepsilon _k w_k\right) \\&\quad \times \prod _{1 \le i \le j \le k} f\left( \varepsilon _i w_i + \varepsilon _j w_j\right) , \end{aligned} \end{aligned}$$

and letting $\alpha _1,\ldots ,\alpha _k \rightarrow 0$, we obtain

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} L(\tfrac{1}{2},\chi _P)^k \sim \sum _{P\in \mathcal {P}_{2g+1,q}} \frac{1}{(\log q)^{k(k+1)/2}} \left( \frac{1}{2} \log |P|\right) ^{\frac{k(k+1)}{2}}\\&\qquad \qquad \times A(\tfrac{1}{2};0,\ldots ,0) 2^{\frac{k(k+1)}{2}} \prod _{i=1}^{k} \frac{i!}{\left( 2i\right) !}, \end{aligned} \end{aligned}$$

(4.32)

as $g\rightarrow \infty$. Summing over P, we get that

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} L(\tfrac{1}{2};\chi _P)^k \sim \sum _{P\in \mathcal {P}_{2g+1,q}} \left( \log _q|P|\right) ^{\frac{k(k+1)}{2}} A_k(\tfrac{1}{2};0,\ldots ,0) \prod _{i=1}^k\frac{i!}{\left( 2i\right) !}\\&\quad = |P| \left( \log _q|P|\right) ^{\frac{k(k+1)}{2}-1} A_k(\tfrac{1}{2};0,\ldots ,0) \prod _{i=1}^k\frac{i!}{\left( 2i\right) !}. \end{aligned} \end{aligned}$$

(4.33)

Hence, we have proved the following.

Theorem 4.2

Conditional on Conjecture 2.2, we have that as $g\rightarrow \infty$, the following holds:

$$\begin{aligned} \begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} L(\tfrac{1}{2},\chi _P)^k \sim |P| \left( \log _q|P|\right) ^{\frac{k(k+1)}{2}-1} A(\tfrac{1}{2};0,\ldots ,0) \prod _{i=1}^{k} \frac{i!}{\left( 2i\right) !}. \end{aligned} \end{aligned}$$

(4.34)

4.4.1 Some conjectural values for leading order asymptotic for the moments of $L(s,\chi _P)$

We end this section by writing the asymptotic formula for the fourth and the fifth moment for our family of L-functions. Theorem 4.2 implies that the leading order for the fourth moment can be written as follows:

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _P\right) ^4 \sim |P| \left( \log _q|P|\right) ^{9}A\left( \tfrac{1}{2};0,0,0,0\right) \prod _{i=1}^4\frac{i!}{\left( 2i\right) !}\\&\quad = \frac{1}{4838400}|P| \left( \log _q|P|\right) ^{9}A\left( \tfrac{1}{2};0,0,0,0\right) , \end{aligned} \end{aligned}$$

(4.35)

where

$$\begin{aligned} \begin{aligned}&A \left( \tfrac{1}{2};0,0,0,0\right) \\&\quad = \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \left( 1- \frac{20 |P|^6-64 |P|^5+90 |P|^4-64 |P|^3+20 |P|^2-1}{|P|^{8}}\right) , \end{aligned} \end{aligned}$$

and the leading order for the fifth moment is

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} L\left( \tfrac{1}{2},\chi _P\right) ^5 \\&\quad \sim |P| \left( \log _q|P|\right) ^{14}A\left( \tfrac{1}{2};0,0,0,0,0\right) \prod _{i=1}^5\frac{i!}{\left( 2i\right) !}\\&\quad = \frac{1}{146313216000}|P| \left( \log _q|P|\right) ^{14}\prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \left( 1-\frac{h(|P|)}{|P|^{12}}\right) . \end{aligned} \end{aligned}$$

(4.36)

with

$$\begin{aligned} \begin{aligned} h(x)&= 50 x^{10}-280 x^9+765 x^8-1248 x^7+1260 x^6 -720 x^5\\&\quad +\,105 x^4 +160 x^3-126 x^2+40 x-5. \end{aligned} \end{aligned}$$

(4.37)

5 Ratios conjecture for L-functions over function fields

The main aim of this section is to obtain a conjectural asymptotic formula for

$$\begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} \frac{\prod _{k=1}^K L(\frac{1}{2}+\alpha _k,\chi _P)}{\prod _{q=1}^Q L(\frac{1}{2}+\gamma _q,\chi _P)}, \end{aligned}$$

(5.1)

where $\mathcal {P}_{2g+1,q}=\{P \text { monic, } P \text { irreducible, deg}(P)=2g+1, P\in \mathbb {F}_{q}[T]\},$ and $\mathfrak {P}=\{L(s,\chi _P):P\in \mathcal {P}_{2g+1,q}\}$. We adapt the original recipe of Conrey, Farmer and Zirnbauer [8] for this family of L-functions.

The idea is to replace the L-functions in the numerator by their “approximate” functional equation:

$$\begin{aligned} L(s,\chi _P)=\sum _{\begin{array}{c} n \text { monic}\\ \text {deg}(n)\le g \end{array}} \frac{\chi _P(n)}{|n|^s}+ \mathcal {X}_P(s) \sum _{\begin{array}{c} n \text { monic}\\ \text {deg}(n)\le g-1 \end{array}} \frac{\chi _P(n)}{|n|^{1-s}}, \end{aligned}$$

(5.2)

and expand the L-functions in the denominator into the series

$$\begin{aligned} \begin{aligned} \frac{1}{L(s,\chi _P)} =&\prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}}\left( 1-\frac{\chi _P(P)}{|P|^s}\right) =&\sum _{n \text { monic}} \frac{\mu (n) \chi _P(n)}{|n|^s}, \end{aligned} \end{aligned}$$

(5.3)

where $\mu (n)$ and $\chi _P(n)$ are defined in Sect. 2.

As in the previous section, we apply the recipe to the quantity

$$\begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} \frac{\prod _{k=1}^K Z_\mathcal {L}(\frac{1}{2}+\alpha _k,\chi _P)}{\prod _{q=1}^Q L(\frac{1}{2}+\gamma _q,\chi _P)}, \end{aligned}$$

(5.4)

where $Z_{\mathcal {L}}(s,\chi _P)$ is defined in (3.7) with “approximate” functional equation given by (3.11). Now expanding the denominator, we get

$$\begin{aligned} \begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}}&\frac{\prod _{k=1}^K Z_\mathcal {L}(\frac{1}{2}+\alpha _k,\chi _P)}{\prod _{q=1}^Q L(\frac{1}{2}+\gamma _q,\chi _P)}\\ =&\sum _{P\in \mathcal {P}_{2g+1,q}} \prod _{k=1}^K Z_\mathcal {L}(\tfrac{1}{2}+\alpha _k,\chi _P) \sum _{\begin{array}{c} h_1,\ldots ,h_Q \\ h_q \text { monic} \end{array}} \frac{\mu (h_1)\cdots \mu (h_Q) \chi _P(h_1 \cdots h_Q)}{|h_1|^{\frac{1}{2}+\gamma _1} \cdots |h_Q|^{\frac{1}{2}+\gamma _Q}}. \end{aligned} \end{aligned}$$

(5.5)

Making use of the “approximate” functional equation (5.2), we have

$$\begin{aligned} \begin{aligned} \prod _{k=1}^K&Z_\mathcal {L}(\tfrac{1}{2}+\alpha _k,\chi _P)\\ =&\sum _{\varepsilon _k\in \{-1,1\}^K} \prod _{k=1}^K \mathcal {X}_P(\tfrac{1}{2}+\varepsilon _k\alpha _k)^{-\frac{1}{2}} \sum _{\begin{array}{c} m_1,\ldots ,m_K \\ m_i \text { monic} \end{array}} \frac{\chi _P(m_1 \cdots m_K)}{|m_1|^{\frac{1}{2}+\varepsilon _1\alpha _1} \cdots |m_K|^{\frac{1}{2}+\varepsilon _K\alpha _K}}, \end{aligned} \end{aligned}$$

(5.6)

so we can write (5.5) as follows:

$$\begin{aligned} \begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}}&\frac{\prod _{k=1}^K Z_\mathcal {L}(\frac{1}{2}+\alpha _k,\chi _P)}{\prod _{q=1}^Q L(\frac{1}{2}+\gamma _q,\chi _P)}\\ =&\sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\varepsilon _k\in \{-1,1\}^K} \prod _{k=1}^K \mathcal {X}_P(\tfrac{1}{2}+\varepsilon _k\alpha _k)^{-\frac{1}{2}}\\&\quad \times \sum _{\begin{array}{c} m_1,\cdots ,m_K \\ h_1,\cdots ,h_Q \\ m_i,h_j \text { monic} \end{array}} \frac{\prod _{q=1}^Q \mu (h_q) \chi _P(\prod _{k=1}^K m_k \prod _{q=1}^Q h_q)}{\prod _{k=1}^K |m_k|^{\frac{1}{2}+\varepsilon _k\alpha _k} \prod _{q=1}^Q |h_q|^{\frac{1}{2}+\gamma _q}}.\\ \end{aligned} \end{aligned}$$

(5.7)

Following the recipe, we replace each summand by its expected value when averaged over primes $P\in \mathcal {P}_{2g+1,q}$, in other words, we have that

$$\begin{aligned} \begin{aligned}&\lim _{\text {deg}(P)\rightarrow \infty } \Bigg (\frac{1}{\#\mathcal {P}_{2g+1,q}} \sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\varepsilon _k\in \{-1,1\}^K} \prod _{k=1}^K \mathcal {X}_P(\tfrac{1}{2}+\varepsilon _k\alpha _k)^{-\frac{1}{2}}\\&\quad \quad \times \sum _{\begin{array}{c} m_1,\ldots ,m_K \\ h_1,\ldots ,h_Q \\ m_i,h_j \text { monic} \end{array}} \frac{\prod _{q=1}^Q \mu (h_q) \chi _P(\prod _{k=1}^K m_k \prod _{q=1}^Q h_q)}{\prod _{k=1}^K |m_k|^{\frac{1}{2}+\varepsilon _k\alpha _k} \prod _{q=1}^Q |h_q|^{\frac{1}{2}+\gamma _q}}\Bigg )\\&\quad = \sum _{\varepsilon _k\in \{-1,1\}^K} \prod _{k=1}^K \mathcal {X}_P (\tfrac{1}{2}+\varepsilon _k\alpha _k)^{-\frac{1}{2}}\\&\quad \quad \times \sum _{\begin{array}{c} m_1,\ldots ,m_K \\ h_1,\ldots ,h_Q \\ m_i,h_j \text { monic} \end{array}} \frac{\prod _{q=1}^Q \mu (h_q) \delta \left( \prod _{k=1}^K m_k \prod _{q=1}^Q h_q\right) }{\prod _{k=1}^K |m_k|^{\frac{1}{2}+\varepsilon _k\alpha _k} \prod _{q=1}^Q |h_q|^{\frac{1}{2}+\gamma _q}}, \end{aligned} \end{aligned}$$

(5.8)

where $\delta (n)=1$ if n is a square and 0 otherwise.

Next we factor out the zeta-function factors. Note that, the main difficulty here is to identify and factor out the appropriate zeta-functions factors that contribute to poles and zeros. With the same notation used in [1], we define the following series:

$$\begin{aligned} G_{\mathfrak {P}} (\alpha ;\gamma ) = \sum _{\begin{array}{c} m_1,\ldots ,m_K \\ h_1,\ldots ,h_Q \\ m_i,h_j \text { monic} \end{array}} \frac{\prod _{q=1}^Q \mu (h_q) \delta \left( \prod _{k=1}^K m_k \prod _{q=1}^Q h_q\right) }{\prod _{k=1}^K |m_k|^{\frac{1}{2}+\varepsilon _k\alpha _k} \prod _{q=1}^Q |h_q|^{\frac{1}{2}+\gamma _q}}. \end{aligned}$$

(5.9)

If $m_k=\prod _P P^{a_k}$ and $h_q=\prod _P P^{c_q}$, then we can write $G_{\mathfrak {P}}(\alpha ;\gamma )$ as a convergent Euler product provided that $\mathfrak {R}(\alpha _k)>0$ and $\mathfrak {R}(\gamma _q)>0$,

$$\begin{aligned} \begin{aligned}&G_{\mathfrak {P}} (\alpha ;\gamma )\\&\quad = \prod _{\begin{array}{c} P \text { monic}\\ \text {irreducible} \end{array}}\left( 1+ \sum _{0<\sum _ka_k+\sum _qc_q \text { is even}} \frac{\prod _{q=1}^Q \mu (P^{c_q})}{ |P|^{\sum _ka_k(\frac{1}{2}+\alpha _k)+\sum _qc_q(\frac{1}{2}+\gamma _q)}}\right) . \end{aligned} \end{aligned}$$

(5.10)

We now write $G_{\mathfrak {P}}$ in terms of the zeta-function of $\mathbb {F}_{q}[T]$. First, we express the contribution of all poles and zeros of (5.10) in terms of $\zeta _A(s)$ by rewriting the Euler product in (5.10) as follows:

$$\begin{aligned} \begin{aligned} G_{\mathfrak {P}} (\alpha ;\gamma )&= \prod _{\begin{array}{c} P \text { monic}\\ \text {irreducible} \end{array}}\Bigg (1+ \sum _{\begin{array}{c} j,k\\ j<k \end{array}} \frac{1}{ |P|^{(\frac{1}{2}+\alpha _j)+(\frac{1}{2}+\alpha _k)}}+ \sum _{k} \frac{1}{ |P|^{(1+2\alpha _k)}}\\&\quad +\, \sum _{\begin{array}{c} r,q\\ r<q \end{array}} \frac{\mu (P)^2}{ |P|^{(\frac{1}{2}+\gamma _r)+(\frac{1}{2}+\gamma _q)}}+ \sum _{k}\sum _q \frac{\mu (P)}{ |P|^{(\frac{1}{2}+\alpha _k)+(\frac{1}{2}+\gamma _q)}}+ \cdots \Bigg ), \end{aligned} \end{aligned}$$

(5.11)

where $\cdots$ are referring to the convergent terms. Recall that

$$\begin{aligned} \begin{aligned} \zeta _A(s)= \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \left( 1-\frac{1}{|P|^s}\right) ^{-1} = \prod _{\begin{array}{c} P \text { monic} \\ \text {irreducible} \end{array}} \left( \sum _{j=0}^\infty \left( \frac{1}{|P|^s}\right) ^j\right) . \end{aligned} \end{aligned}$$

(5.12)

We can see from (5.11) that the terms with $\sum _{k=1}^Ka_k+\sum _{q=1}^Q c_q=2$ contribute to the poles and zeros. The poles are coming from the terms with $a_j=a_k=1,1\le j<k\le K,$ $a_k=2, 1\le k\le K,$ and also from the terms with $c_r=c_q=1, 1\le r < q\le Q$. Note that there are no poles coming from the terms with $c_q=2, 1\le q \le Q,$ since $\mu (P^2)=0.$ Moreover, the zeros come from the terms with $a_k=c_q=1$ with $1\le k\le K,$ and $1\le q\le Q.$

From the above, we can define the function $Y_{\mathfrak {P}}(\alpha ;\gamma )$ in terms of $\zeta _A(s)$ by

$$\begin{aligned} Y_{\mathfrak {P}}(\alpha ;\gamma ):= \frac{\prod _{1\le j\le k\le K}\zeta _A(1+\alpha _j+\alpha _k)\prod _{1\le r\le q\le Q}\zeta _A(1+\gamma _r+\gamma _q)}{\prod _{k=1}^k\prod _{q=1}^Q \zeta _A(1+\alpha _k+\gamma _q)}. \end{aligned}$$

(5.13)

Thus, we can factor out $Y_{\mathfrak {P}}(\alpha ;\gamma )$ from $G_{\mathfrak {P}}(\alpha ;\gamma )$, such that

$$\begin{aligned} G_{\mathfrak {P}}(\alpha ;\gamma )= Y_{\mathfrak {P}}(\alpha ;\gamma ) A_{\mathfrak {P}}(\alpha ;\gamma ), \end{aligned}$$

(5.14)

where $A_{\mathfrak {P}}(\alpha ;\gamma )$ is the Euler product that converges absolutely for all of the variables in the small discs around 0:

$$\begin{aligned} \begin{aligned}&A_{\mathfrak {P}}(\alpha ;\gamma )\\&\quad = \prod _{\begin{array}{c} P \text { monic}\\ \text {irreducible} \end{array}} \frac{\prod _{1\le j\le k\le K} \left( 1-\frac{1}{|P|^{1+\alpha _j+\alpha _k}}\right) \prod _{1\le r\le q\le Q} \left( 1-\frac{1}{|P|^{1+\gamma _r+\gamma _q}}\right) }{\prod _{k=1}^k\prod _{q=1}^Q \left( 1-\frac{1}{|P|^{1+\alpha _k+\gamma _q}}\right) }\\&\quad \quad \times \left( 1+ \sum _{0<\sum _ka_k+\sum _qc_q \text { is even}} \frac{\prod _{q=1}^Q \mu (P^{c_q})}{ |P|^{\sum _ka_k(\frac{1}{2}+\alpha _k)+\sum _qc_q(\frac{1}{2}+\gamma _q)}}\right) . \end{aligned} \end{aligned}$$

(5.15)

Returning to the recipe, we can conclude from (5.7), (5.9), and (5.14) that

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \frac{\prod _{k=1}^K Z_\mathcal {L}(\frac{1}{2}+\alpha _k,\chi _P)}{\prod _{q=1}^Q L(\frac{1}{2}+\gamma _q,\chi _P)}\\&\quad = \sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\varepsilon _k\in \{-1,1\}^K} \prod _{k=1}^K \mathcal {X}_P\left( \tfrac{1}{2}+\varepsilon _k\alpha _k\right) ^{-\frac{1}{2}} Y_{\mathfrak {P}}(\varepsilon _1\alpha _1,\cdots ,\varepsilon _k\alpha _k;\gamma )\\&\quad \quad \times A_{\mathfrak {P}}(\varepsilon _1\alpha _1,\cdots ,\varepsilon _k\alpha _k;\gamma ) +o\left( |P|\right) ,\\ \end{aligned} \end{aligned}$$

(5.16)

Now, using (3.7), we have

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \frac{\prod _{k=1}^K L(\frac{1}{2}+\alpha _k,\chi _P)}{\prod _{q=1}^Q L(\frac{1}{2}+\gamma _q,\chi _P)}\\&\quad = \sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\varepsilon _k\in \{-1,1\}^K} \prod _{k=1}^K \mathcal {X}_P\left( \tfrac{1}{2}+\alpha _k\right) ^{\frac{1}{2}} \mathcal {X}_P\left( \tfrac{1}{2}+\alpha _k\right) ^{-\frac{1}{2}}\\&\quad \quad \times Y_{\mathfrak {P}}(\varepsilon _1\alpha _1,\ldots ,\varepsilon _k\alpha _k;\gamma ) A_{\mathfrak {P}}(\varepsilon _1\alpha _1,\ldots ,\varepsilon _k\alpha _k;\gamma ) +o\left( |P|\right) .\\ \end{aligned} \end{aligned}$$

(5.17)

Remembering that

$$\begin{aligned} \mathcal {X}_P(s) = |P|^{\frac{1}{2}-s} \mathcal {X}(s) \end{aligned}$$

(5.18)

with

$$\begin{aligned} \mathcal {X}(s)= q^{-\frac{1}{2}+s}, \end{aligned}$$

(5.19)

we have that

$$\begin{aligned} \begin{aligned}&\prod _{k=1}^K \mathcal {X}_P\left( \tfrac{1}{2}+\varepsilon _k\alpha _k\right) ^{-\frac{1}{2}} \mathcal {X}_P\left( \tfrac{1}{2}+\varepsilon _k\alpha _k\right) ^{\frac{1}{2}}\\&\quad = |P|^{\frac{1}{2}\sum _{k=1}^K (\varepsilon _k\alpha _k-\alpha _k)} \prod _{k=1}^K \mathcal {X}\left( \tfrac{1}{2}+\tfrac{\alpha _k-\varepsilon _k\alpha _k}{2}\right) . \end{aligned} \end{aligned}$$

(5.20)

For positive real parts of $\alpha _k$ and $\gamma _q$, we have

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \frac{\prod _{k=1}^K L(\frac{1}{2}+\alpha _k,\chi _P)}{\prod _{q=1}^Q L(\frac{1}{2}+\gamma _q,\chi _P)}\\&\quad = \sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\varepsilon _k\in \{-1,1\}^K}|P|^{\frac{1}{2}\sum _{k=1}^K (\varepsilon _k\alpha _k-\alpha _k)} \prod _{k=1}^K \mathcal {X}\left( \tfrac{1}{2}+\tfrac{\alpha _k-\varepsilon _k\alpha _k}{2}\right) \\&\quad \quad \times Y_{\mathfrak {P}}(\varepsilon _1\alpha _1,\ldots ,\varepsilon _k\alpha _k;\gamma ) A_{\mathfrak {P}}(\varepsilon _1\alpha _1,\ldots ,\varepsilon _k\alpha _k;\gamma ) +o\left( |P|\right) .\\ \end{aligned} \end{aligned}$$

(5.21)

Finally, if we let

$$\begin{aligned} \begin{aligned} H_{\mathfrak {P},|P|,\alpha ,\gamma }(w;\gamma ) =&\left| P\right| ^{\frac{1}{2}\sum _{k=1}^k w_k} \prod _{k=1}^K \mathcal {X}\left( \tfrac{1}{2}+\tfrac{\alpha _k-w_k}{2}\right) \\&\times Y_{\mathfrak {P}}(w;\gamma ) A_{\mathfrak {P}}(w;\gamma ), \end{aligned} \end{aligned}$$

(5.22)

then the conjecture may be formulated as follows:

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \frac{\prod _{k=1}^KL(\frac{1}{2}+\alpha _k,\chi _P)}{\prod _{q=1}^QL(\frac{1}{2}+\gamma _q,\chi _P)}\\&\quad =\sum _{P\in \mathcal {P}_{2g+1,q}} |P|^{-\frac{1}{2}\sum _{k=1}^K \alpha _k} \sum _{\varepsilon \in \{-1,1\}^K} H_{\mathfrak {P},|P|,\alpha ,\gamma }(\varepsilon \alpha ;\gamma )+o\left( |P|\right) .\\ \end{aligned} \end{aligned}$$

(5.23)

5.1 Refinements of Conjecture

In this section, we state the final form of our ratios conjecture. In the first part, we derive a closed form expression for the Euler product $A_\mathfrak {P}(\alpha ;\gamma ),$ and in the second part, we express the combinatorial sum as a multiple integral.

5.1.1 Closed form expression for $A_\mathfrak {P}$

Suppose that $f(x)=1+\sum _{n=1}^\infty u_nx^n$, then

$$\begin{aligned} \begin{aligned} \sum _{n \text { even}} u_n x^n&=\frac{1}{2} \left( f(x)+f(-x)-2\right) , \end{aligned} \end{aligned}$$

(5.24)

and so, let

$$\begin{aligned} \begin{aligned} f\left( \frac{1}{|P|}\right) =&\sum _{a_k,c_q} \frac{\prod _{q=1}^Q\mu (P^{c_q})}{|P|^{\sum _ka_k(\frac{1}{2}+\alpha _k)+\sum _qc_q(\frac{1}{2}+\gamma _q)}}\\ =&\sum _{a_k} \prod _{k=1}^K\frac{1}{|P|^{a_k(\frac{1}{2}+\alpha _k)}} \sum _{c_q} \prod _{q=1}^Q\frac{\mu (P^{c_q})}{|P|^{c_q(\frac{1}{2}+\gamma _q)}}\\ =&\frac{\prod _{q=1}^Q\left( 1-\frac{1}{|P|^{\frac{1}{2}+\gamma _q}}\right) }{\prod _{k=1}^K\left( 1-\frac{1}{|P|^{\frac{1}{2}+\alpha _k}}\right) }. \end{aligned} \end{aligned}$$

(5.25)

Using the above equations, we can establish the following lemma.

Lemma 5.1

We have that,

$$\begin{aligned} \begin{aligned}&1+ \sum _{\sum _ka_k+\sum _qc_q \text { even}} \frac{\prod _{q=1}^Q\mu (P^{c_q})}{|P|^{\sum _ka_k(\frac{1}{2}+\alpha _k)+\sum _qc_q(\frac{1}{2}+\gamma _q)}}\\&\quad = \frac{1}{2} \left( \frac{\prod _{q=1}^Q\left( 1-\frac{1}{|P|^{\frac{1}{2}+\gamma _q}}\right) }{\prod _{k=1}^K\left( 1-\frac{1}{|P|^{\frac{1}{2}+\alpha _k}}\right) }+\frac{\prod _{q=1}^Q\left( 1+\frac{1}{|P|^{\frac{1}{2}+\gamma _q}}\right) }{\prod _{k=1}^K\left( 1+\frac{1}{|P|^{\frac{1}{2}+\alpha _k}}\right) }\right) . \end{aligned} \end{aligned}$$

(5.26)

The following result is a direct corollary from Lemma 5.1 and Eq. (5.15).

Corollary 5.2

$$\begin{aligned} \begin{aligned} A_{\mathfrak {P}}(\alpha ;\gamma )&= \prod _{\begin{array}{c} P \text { monic}\\ \text {irreducible} \end{array}} \frac{\prod _{1\le j\le k\le K} \left( 1-\frac{1}{|P|^{1+\alpha _j+\alpha _k}}\right) \prod _{1\le r\le q\le Q} \left( 1-\frac{1}{|P|^{1+\gamma _r+\gamma _q}}\right) }{\prod _{k=1}^k\prod _{q=1}^Q \left( 1-\frac{1}{|P|^{1+\alpha _k+\gamma _q}}\right) }\\&\quad \times \left( \frac{1}{2} \left( \frac{\prod _{q=1}^Q\left( 1-\frac{1}{|P|^{\frac{1}{2}+\gamma _q}}\right) }{\prod _{k=1}^K\left( 1-\frac{1}{|P|^{\frac{1}{2}+\alpha _k}}\right) }+\frac{\prod _{q=1}^Q\left( 1+\frac{1}{|P|^{\frac{1}{2}+\gamma _q}}\right) }{\prod _{k=1}^K\left( 1+\frac{1}{|P|^{\frac{1}{2}+\alpha _k}}\right) }\right) \right) . \end{aligned} \end{aligned}$$

(5.27)

5.1.2 The final form of the ratios conjecture

To obtain our final form of the Ratios Conjecture 2.3, we need the following lemma (Lemma 6.8, [8]).

Lemma 5.3

Suppose that $F(z)=F(z_1,\cdots ,z_K)$ is a function of K variables, which is symmetric and regular near $(0,\cdots ,0).$ Suppose further that f(s) has a simple pole of residue 1 at $s=0$ but is otherwise analytic in $|s|\le 1.$ Let either

$$\begin{aligned} H(z_1,\ldots ,z_K)=F(z_1,\ldots ,z_K)\prod _{1\le j\le k\le K} f(z_j+z_k) \end{aligned}$$

(5.28)

or

$$\begin{aligned} H(z_1,\ldots ,z_K)=F(z_1,\ldots ,z_K)\prod _{1\le j< k\le K} f(z_j+z_k). \end{aligned}$$

(5.29)

If $|\alpha _k|<1$, then

$$\begin{aligned} \begin{aligned}&\sum _{\varepsilon \in \{-1,1\}^K} H(\varepsilon _1\alpha _1,\ldots ,\varepsilon _K\alpha _K)\\&\quad = \frac{(-1)^{K(K-1)/2}2^K}{K! (2\pi i)^K} \int _{|z_i|=1} \frac{H(z_1,\ldots ,z_K) \Delta (z_1^2,\ldots ,z_K^2)^2 \prod _{k=1}^Kz_k}{\prod _{j=1}^K\prod _{k=1}^K(z_k-\alpha _j)(z_k+\alpha _j)} \mathrm{{d}}z_1\cdots \mathrm{{d}}z_K \end{aligned} \end{aligned}$$

(5.30)

and

$$\begin{aligned} \begin{aligned}&\sum _{\varepsilon \in \{-1,1\}^K} \text {sgn}(\varepsilon ) H(\varepsilon _1\alpha _1,\ldots ,\varepsilon _K\alpha _K)\\&\quad = \frac{(-1)^{K(K-1)/2}2^K}{K! (2\pi i)^K} \int _{|z_i|=1} \frac{H(z_1,\ldots ,z_K) \Delta (z_1^2,\ldots ,z_K^2)^2 \prod _{k=1}^K\alpha _k}{\prod _{j=1}^K\prod _{k=1}^K(z_k-\alpha _j)(z_k+\alpha _j)} \mathrm{{d}}z_1\cdots \mathrm{{d}}z_K. \end{aligned} \end{aligned}$$

(5.31)

Now, we are in a position to present the final form of the ratios conjecture 2.3.

Conjecture 5.4

Suppose that the real parts of $\alpha _k$ and $\gamma _q$ are positive. Then we have,

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}}\frac{\prod _{k=1}^KL\left( \frac{1}{2}+\alpha _k,\chi _P\right) }{\prod _{q=1}^QL\left( \frac{1}{2}+\gamma _q,\chi _P\right) }\\&\quad =\sum _{P\in \mathcal {P}_{2g+1,q}} |P|^{-\frac{1}{2}\sum _{k=1}^K \alpha _k} \frac{(-1)^{K(K-1)/2}2^K}{K! (2\pi i)^K}\\&\quad \quad \times \int _{|z_i|=1} \frac{H_{\mathfrak {P},|P|,\alpha ,\gamma }(z_1,\ldots ,z_K) \Delta (z_1^2,\ldots ,z_K^2)^2 \prod _{k=1}^Kz_k}{\prod _{j=1}^K\prod _{k=1}^K(z_k-\alpha _j)(z_k+\alpha _j)} \mathrm{{d}}z_1\cdots \mathrm{{d}}z_K\\&\quad \quad +o(|P|). \end{aligned} \end{aligned}$$

(5.32)

6 One-level density

In this section, we give an application of the Ratios Conjecture 2.3 for L-functions over function fields. We compute a smooth linear statistic, the one-level density for the family of quadratic Dirichlet L-functions associated with monic irreducible polynomials in $\mathbb {F}_{q}[T]$. The one-level density for the family of quadratic Dirichlet L-functions over fundamental discriminants was computed using the rations conjecture by Conrey and Snaith [15] in the number field setting and by Andrade and Keating [4] in the function field setting.

Consider

$$\begin{aligned} R_P(\alpha ;\gamma )= \sum _{P\in \mathcal {P}_{2g+1,q}} \frac{L(\frac{1}{2}+\alpha ,\chi _P)}{L(\frac{1}{2}+\gamma ,\chi _P)}. \end{aligned}$$

(6.1)

Using the ratios conjecture as presented in the last section with one L-function in the numerator and one L-function in the denominator, we arrive at the following particular conjecture.

Conjecture 6.1

With $-\frac{1}{4}<\mathfrak {R}(\alpha )<\frac{1}{4}, \frac{1}{\log |P|}\ll \mathfrak {R}(\gamma )<\frac{1}{4}$ and $\mathfrak {I}(\alpha ),\mathfrak {I}(\gamma )\ll _\epsilon |P|^{1-\epsilon }$ for every $\epsilon >0,$ we have

$$\begin{aligned} \begin{aligned} R_P(\alpha ;\gamma )=&\sum _{P\in \mathcal {P}_{2g+1,q}}\frac{L(\frac{1}{2}+\alpha ,\chi _P)}{L(\frac{1}{2}+\gamma ,\chi _P)}\\ =&\sum _{P\in \mathcal {P}_{2g+1,q}} \Bigg ( \frac{\zeta _A(1+2\alpha )}{\zeta _A(1+\alpha +\gamma )} + |P|^{-\alpha } \mathcal {X}\left( \tfrac{1}{2}+\alpha \right) \\&\times \frac{\zeta _A(1-2\alpha )}{\zeta _A(1-\alpha +\gamma )}\Bigg )+o\left( |P|\right) . \end{aligned} \end{aligned}$$

(6.2)

To compute the one-level density, we need to have a formula for

$$\begin{aligned} \begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}} \frac{L'(\frac{1}{2}+r,\chi _P)}{L(\frac{1}{2}+r,\chi _P)} =&\frac{\mathrm{{d}}}{\mathrm{{d}}\alpha } R_\mathcal {P}(\alpha ;\gamma )\Big |_{\alpha =\gamma =r}. \end{aligned} \end{aligned}$$

(6.3)

A direct calculation gives

$$\begin{aligned} \begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}\alpha }\left( \frac{\zeta _A(1+2\alpha )}{\zeta _A(1+\alpha +\gamma )}\right) \Bigg |_{\alpha =\gamma =r}=\frac{\zeta _A'(1+2r)}{\zeta _A(1+2r)} \end{aligned} \end{aligned}$$

(6.4)

and that

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{{d}}}{\mathrm{{d}}\alpha }\Big (|P|^{-\alpha } \mathcal {X}(\tfrac{1}{2}+\alpha ) \frac{\zeta _A(1-2\alpha )}{\zeta _A(1-\alpha +\gamma )}\Big )\Bigg |_{\alpha =\gamma =r}\\&\quad = - \left( \log q\right) |P|^{-r} \mathcal {X}\left( \tfrac{1}{2}+r\right) \zeta _A(1-2r).\\ \end{aligned} \end{aligned}$$

(6.5)

Therefore, the ratios conjecture implies that the following result holds.

Theorem 6.2

Assuming Conjecture 6.1, $\frac{1}{\log |P|}\ll \mathfrak {R}(r)<\frac{1}{4}$ and $\mathfrak {I}(r)\ll _\epsilon |P|^{1-\epsilon }$ for every $\epsilon >0,$ we have

$$\begin{aligned} \begin{aligned} \sum _{P\in \mathcal {P}_{2g+1,q}}&\frac{L'(\frac{1}{2}+r,\chi _P)}{L(\frac{1}{2}+r,\chi _P)}\\ =&\sum _{P\in \mathcal {P}_{2g+1,q}} \Big ( \frac{\zeta _A'(1+2r)}{\zeta _A(1+2r)}- \left( \log q\right) |P|^{-r} \mathcal {X}\left( \tfrac{1}{2}+r\right) \\&\quad \times \zeta _A(1-2r)\Big ) +o\left( |P|\right) . \end{aligned} \end{aligned}$$

(6.6)

We have available all the necessary machinery to derive the formula for the one-level density for the zeros of Dirichlet L-functions associated to quadratic characters $\chi _P$ with $P\in \mathcal {P}_{2g+1,q}$, complete with lower-order terms.

Let $\gamma _P$ be the ordinate of a generic zero of $L(s,\chi _P)$ on the half-line. Since $L(s,\chi _P)$ is a function of $u=q^{-s}$ and periodic with period $2\pi i/\log q$, we can restrict our analysis of the zeros for the range $-\pi i/\log q \le \mathfrak {I}(s) \le \pi i/\log q.$ Consider the one-level density

$$\begin{aligned} S_1(f):=\sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\gamma _P} f(\gamma _P), \end{aligned}$$

(6.7)

where f is an even $2\pi /\log q$-periodic test functions and holomorphic.

Using Cauchy’s Theorem, we have

$$\begin{aligned} S_1(f)=\sum _{P\in \mathcal {P}_{2g+1,q}} \frac{1}{2\pi i} \left( \int _{(c)}-\int _{(1-c)}\right) \frac{L'(s,\chi _P)}{L(s,\chi _P)} f\left( -i\left( s-1/2\right) \right) \mathrm{{d}}s, \end{aligned}$$

(6.8)

where (c) is the vertical line from $c-\pi i/\log q$ to $c+\pi i/\log q$ and $1/2+1/\log |P|<c<3/4$. For the integral on the (c)-line, we make the following variable change, letting $s\rightarrow c+it$, so

$$\begin{aligned} \begin{aligned} \frac{1}{2\pi }\int _{-\pi /\log q}^{\pi /\log q}f(-i(it+c-1/2)) \sum _{P\in \mathcal {P}_{2g+1,q}} \frac{L'(c+it,\chi _P)}{L(c+it,\chi _P)}\mathrm{{d}}t. \end{aligned} \end{aligned}$$

(6.9)

Since the integrand is regular at $t=0$, we move the path of the integration to $c=1/2$ and replace the sum over P by Theorem 6.2 to obtain

$$\begin{aligned} \begin{aligned}&\frac{1}{2\pi }\int _{-\pi /\log q}^{\pi /\log q} f(t) \sum _{P\in \mathcal {P}_{2g+1,q}} \Bigg ( \frac{\zeta _A'(1+2it)}{\zeta _A(1+2it)}\\&\quad -\, \left( \log q\right) |P|^{-it} \mathcal {X}\left( \frac{1}{2}+it\right) \zeta _A(1-2it)\Bigg )\mathrm{{d}}t + o\left( |P|\right) . \end{aligned} \end{aligned}$$

(6.10)

The functional equation (2.10) implies that

$$\begin{aligned} \begin{aligned} \frac{L'(1-s,\chi _P)}{L(1-s,\chi _P)}= \frac{\mathcal {X}_P'(s)}{\mathcal {X}_P(s)}-\frac{L'(s,\chi _P)}{L(s,\chi _P)} \end{aligned} \end{aligned}$$

(6.11)

with

$$\begin{aligned} \frac{\mathcal {X}_P'(s)}{\mathcal {X}_P(s)}= - \log |P|+\frac{\mathcal {X}'(s)}{\mathcal {X}(s)}. \end{aligned}$$

(6.12)

For the integral on the $(1-c)$-line, we change variables, letting $s\rightarrow 1-s$, then use (6.11) and with the similar calculations as for the integral on the (c)-line, we obtain the following theorem.

Theorem 6.3

Assuming the ratios Conjecture 6.1, we have that

$$\begin{aligned} \begin{aligned} S_1(f)=&\sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\gamma _P} f(\gamma _P)\\ =&\frac{1}{2\pi } \int _{-\pi /\log q}^{\pi /\log q} f(t) \sum _{P\in \mathcal {P}_{2g+1,q}} \Bigg ( \log |P|+\frac{\mathcal {X}'(\frac{1}{2}-it)}{\mathcal {X}(\frac{1}{2}-it)} \\&+ 2 \Bigg ( \frac{\zeta _A'(1+2it)}{\zeta _A(1+2it)}- \left( \log q\right) |P|^{-it} \mathcal {X}\left( \tfrac{1}{2}+it\right) \zeta _A(1-2it)\Bigg ) \Bigg )\mathrm{{d}}t \\&+ o\left( |P|\right) , \end{aligned} \end{aligned}$$

(6.13)

where $\gamma _P$ is the ordinate of a generic zero of $L(s,\chi _P)$ and f is an even and periodic suitable test function.

6.1 The scaled one-level density

Defining

$$\begin{aligned} f(t)= h\left( \frac{t(2g\log q)}{2\pi }\right) \end{aligned}$$

(6.14)

and scaling the variable t from Theorem 6.3 as

$$\begin{aligned} \tau =\frac{t(2g\log q)}{2\pi }, \end{aligned}$$

(6.15)

we have that

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\gamma _P} f\left( \gamma _P \frac{2g\log q}{2\pi }\right) \\&\quad = \frac{1}{2g \log q} \int _{-g}^{g} h(\tau ) \sum _{P\in \mathcal {P}_{2g+1,q}} \Bigg ( \log |P|+\frac{\mathcal {X}'\left( \frac{1}{2}-\frac{2\pi i \tau }{2g\log q}\right) }{\mathcal {X}\left( \frac{1}{2}-\frac{2\pi i \tau }{2g\log q}\right) } \\&\quad \quad + 2 \Bigg ( \frac{\zeta _A'\left( 1+\frac{4\pi i\tau }{2g\log q}\right) }{\zeta _A\left( 1+\frac{4\pi i\tau }{2g\log q}\right) }- \left( \log q\right) e^{(-2\pi i \tau /2g \log q)\log |P|} \mathcal {X}\left( \tfrac{1}{2}+\tfrac{2\pi i \tau }{2g\log q}\right) \\&\quad \quad \times \zeta _A\left( 1-\frac{4\pi i\tau }{2g\log q}\right) \Bigg ) \Bigg )d\tau + o\left( |P|\right) . \end{aligned} \end{aligned}$$

(6.16)

Writing

$$\begin{aligned} \begin{aligned} \zeta _A(1+s)= \frac{1}{s \log q} + \frac{1}{2} + \frac{1}{12} (\log q) s + O(s^2), \end{aligned} \end{aligned}$$

(6.17)

and

$$\begin{aligned} \begin{aligned} \frac{\zeta _A'(1+s)}{\zeta _A(1+s)}= -s^{-1} + \frac{1}{2}\log q - \frac{1}{12} (\log q)^2 s + O(s^3), \end{aligned} \end{aligned}$$

(6.18)

we have

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\gamma _P} f\left( \gamma _P \frac{2g\log q}{2\pi }\right) \\&\quad = \frac{1}{2g \log q} \int _{-g}^{g} h(\tau ) \sum _{P\in \mathcal {P}_{2g+1,q}} \Bigg ( \log |P|+\frac{\mathcal {X}'\left( \frac{1}{2}-\frac{2\pi i \tau }{2g\log q}\right) }{\mathcal {X}\left( \frac{1}{2}-\frac{2\pi i \tau }{2g\log q}\right) } \\&\quad \quad + 2 \Bigg ( -\frac{2g\log q}{4\pi i\tau } + \frac{1}{2}\log q - \frac{1}{12} (\log q) \frac{4\pi i\tau }{2g} - \left( \log q\right) e^{(-2\pi i \tau /2g \log q)\log |P|} \\&\quad \quad \times \mathcal {X}\left( \tfrac{1}{2}+\tfrac{2\pi i \tau }{2g\log q}\right) \left( -\frac{2g}{4\pi i \tau } + \frac{1}{2} - \frac{1}{12} \frac{4\pi i\tau }{2g}\right) \Bigg ) \Bigg )\mathrm{{d}}\tau + o\left( |P|\right) . \end{aligned} \end{aligned}$$

(6.19)

then, for g large, only the term $\log |P|$, the $\zeta _{A}^{'}/\zeta _{A}$ and the final term in the integral contribute, yielding the asymptotic

$$\begin{aligned} \begin{aligned}&\sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\gamma _P} f\left( \gamma _P \frac{2g\log q}{2\pi }\right) \\&\quad \sim \, \frac{1}{2g \log q} \int _{-\infty }^{\infty } h(\tau ) \Bigg (\left( \#\mathcal {P}_{2g+1,q}\right) \log |P|\\&\quad \quad - \left( \#\mathcal {P}_{2g+1,q}\right) \frac{2g \log q}{2\pi i \tau } + \left( \#\mathcal {P}_{2g+1,q}\right) e^{-2\pi i \tau } \frac{2g \log q}{2\pi i \tau }\Bigg ) \mathrm{{d}}\tau . \end{aligned} \end{aligned}$$

(6.20)

However, since h is an even function, we can drop out the middle term and the last term can be duplicated with a change of sign of $\tau ,$ leaving

$$\begin{aligned} \begin{aligned}&\lim _{g\rightarrow \infty } \frac{1}{\#\mathcal {P}_{2g+1,q}} \sum _{P\in \mathcal {P}_{2g+1,q}} \sum _{\gamma _P} f\left( \gamma _P \frac{2g\log q}{2\pi }\right) \\&\quad = \, \int _{-\infty }^{\infty } h(\tau ) \Bigg (1 + e^{-2\pi i \tau } \frac{1}{2\pi i \tau }+ e^{2\pi i \tau } \frac{1}{-2\pi i \tau }\Bigg ) \mathrm{{d}}\tau \\&\quad = \, \int _{-\infty }^{\infty } h(\tau ) \Bigg (1 + \frac{1}{2\pi \tau }\Big (\left( \cos (2\pi \tau ) -\sin (2\pi \tau )\right) - \left( \cos (2\pi \tau )-\sin (2\pi \tau )\right) \Big )\Bigg ) \mathrm{{d}}\tau \\&\quad = \, \int _{-\infty }^{\infty } h(\tau ) \Bigg (1 + \frac{1}{2\pi \tau }\Big (-2\sin (2\pi \tau ) \Big )\Bigg ) d\tau \\&\quad = \, \int _{-\infty }^{\infty } h(\tau ) \Bigg (1 - \frac{\sin (2\pi \tau )}{\pi \tau }\Bigg ) \mathrm{{d}}\tau . \end{aligned} \end{aligned}$$

(6.21)