Average Values of $L$-series for Real Characters in Function Fields

We establish asymptotic formulae for the first and second moments of quadratic Dirichlet $L$--functions, at the centre of the critical strip, associated to the real quadratic function field $k(\sqrt{P})$ and inert imaginary quadratic function field $k(\sqrt{\gamma P})$ with $P$ being a monic irreducible polynomial over a fixed finite field $\mathbb{F}_{q}$ of odd cardinality $q$ and $\gamma$ a generator of $\mathbb{F}_{q}^{\times}$. We also study mean values for the class number and for the cardinality of the second $K$-group of maximal order of the associated fields for ramified imaginary, real, and inert imaginary quadratic function fields over $\mathbb{F}_{q}$. One of the main novelties of this paper is that we compute the second moment of quadratic Dirichlet $L$-functions associated to monic irreducible polynomials. It is worth noting that the similar second moment over number fields is unknown. The second innovation of this paper comes from the fact that, if the cardinality of the ground field is even then the task of average $L$-functions in function fields is much harder and, in this paper, we are able to handle this strenuous case and establish several mean values results of $L$-functions over function fields.

1. Introduction and some basic facts 1 1. 1. Introduction and some basic facts 1.1. Introduction. It is a profound problem in analytic number theory to understand the distribution of values of L(s, χ p ), the Dirichlet L-functions associated to the quadratic character χ p , for fixed s and variable p, where for a prime number p ≡ v (mod 4) with v = 1 or 3, the quadratic character χ p (n) is defined by the Legendre symbol χ p (n) = ( n p ). The problem about the distribution of values of Dirichlet L-functions with real characters χ modulo a prime p was first studied by Elliott in [9] and later some of his results were generalized by Stankus [18].
For Re(s) > 1 2 , Stankus proved that L(s, χ p ) is in a given Borel set B in the complex plane with a certain probability which depends on s and B. However the same question is non-trivial if we consider s in the center of the critical strip, i.e., Re(s) = 1 2 . In particular, it is a challenging (and open) problem to decide if L( 1 2 , χ p ) = 0 for all quadratic characters χ p . Appears that this fact was first conjectured by Chowla [8].
It is also a difficult problem to determine whether or not L( 1 2 , χ p ) = 0 for infinitely many primes p. A natural strategy to attack this problem is to prove that L( 1 2 , χ p ) has a positive average value when 0 < p ≤ X and X is large. That is, p≤X p≡v (mod 4) L( 1 2 , χ p ) > 0, (1.1) when X is large. In this context, Goldfeld and Viola [10] have conjectured an asymptotic formula for p≤X p≡v (mod 4) L( 1 2 , χ p ), (1.2) and Jutila [13] was able to establish the following asymptotic formula: Theorem 1.1 (Jutila). For v = 1 or 3, we have p≤X p≡v (mod 4) where the implied constant is not effectively calculable. The following estimate is effective: In a recent paper [4], the authors raise the question about higher moments for the family of quadratic Dirichlet L-functions associated to χ p . In other words, the problem is when X → ∞ and k > 1.
The only known asymptotic formulae for (1.5) are those given in the Theorem 1.1, i.e., we have asymptotic formulas merely when k = 1 and it is an important open problem for k > 1.
The first aim of this paper is to study the function field analogue of the problem above in the same spirit as those recent results obtained by Andrade and Keating in [1][2][3][4] and extend their results. The second aim of the paper is to derive asymptotic formulas for the mean values of quadratic Dirichlet L-functions over the rational function field at the special point s = 1 and as an immediate corollary to obtain the mean values of the associated class numbers over function fields.
One of the main novelties of this paper is that we compute the second moment of quadratic Dirichlet L-functions associated to monic irreducible polynomials of even degree (the odd degree case was computed by Andrade-Keating in [4]), and in this way we are able to go beyond of what is known in the number field case. The second innovation of this paper comes from the fact that, if the cardinality of the ground field is even then the task of average L-functions in function fields is much harder and, in this paper, we are able to handle this strenuous case and establish several mean values results of L-functions over function fields. (See next section for more details.) 1.2. Zeta function of curves. Let F q be a finite field of odd cardinality, A = F q [T ] the polynomial ring over F q and we denote by k = F q (T ) the rational function field over F q . We consider C to be any smooth, projective, geometrically connected curve of genus g ≥ 1 defined over the finite field F q . In this setting Artin [5] has defined the zeta function of the curve C as where N n (C) := Card(C(F q )) is the number of points on C with coordinates in a field extension F q n of F q of degree n ≥ 1 and u = q −s . It is well known that the zeta function attached to C is a rational function as proved by Weil [19] and in this case is presented in the following form , (1.7) where L C (u) ∈ Z[u] is a polynomial of degree 2g, called the L-polynomial of the curve C. The Riemann-Roch theorem for function fields (see [16,Theorem 5.4 and Theorem 5.9]) show us that L C (u) satisfies the following functional equation: And the Riemann Hypothesis for curves over finite fields is a theorem in this setting, which was established by Weil [19] in 1948, and it says that the zeros of L C (u) all lie on the circle |u| = q − 1 2 , i.e., (1 − α j u), with |α j | = √ q for all j. (1.9) 1.3. Some Background on A = F q [T ]. Let A + denote the set of monic polynomials in A and P denote the set of monic irreducible polynomials in A. For a positive integer n we denote by A + n to be the set of monic polynomials in A of degree n and by P n to be the set of monic irreducible polynomials in A of degree n. Throughout this paper, a monic irreducible polynomial P ∈ P will be also called a "prime" polynomial. The norm of a polynomial f ∈ A is defined to be |f | := q deg(f ) for f = 0, and |f | = 0 for f = 0. The sign sgn(f ) of f is the leading coefficient of f . The zeta function of A is defined for Re(s) > 1 to be the following infinite series (1.10) It is easy to show (see [16,Chapter 2]) that the zeta function ζ A (s) is a very simple function and can be rewritten as Hoffstein and Rosen [11] were one of the first to study mean values of L-functions over function fields. In their beautiful paper, they established several mean values of L-series over the rational function field they considered averages over all monic polynomials, as well as the sum over square-free polynomials in F q [T ]. But in their paper, they never consider mean values of L-functions associated to monic irreducible polynomials over F q [T ]. In this paper, we investigate the problem of averaging L-functions over prime polynomials and we compute the first and the second moment of several families of L-functions, thus extending the pioneering work of Hoffstein and Rosen. It is also important to note that our methods are totally different from those used by Hoffstein and Rosen and are based on the use of the approximate functional equation for function fields.

2.1.
Odd characteristic case. In this subsection we assume that q is odd. First we present some preliminary facts on quadratic Dirichlet L-functions for the rational function field k = F q (T ) and for this we use Rosen's book [16] as a guide to the notations and definitions. We also present the results of Andrade-Keating [4], which is the main inspiration for this article.
Let H be the set of non-constant square free polynomials D in A with sgn(D) ∈ {1, γ}. Then any quadratic extension K of k can be written uniquely as K = K D := k( √ D) for D ∈ H. The infinite prime ∞ = (1/T ) of k is ramified, splits, or is inert in K D according as deg(D) is odd, deg(D) is even and sgn(D) = 1, or deg(D) is even and sgn(D) = γ. Then K D is called ramified imaginary, real, or inert imaginary, respectively. The genus g D of K D is given by For D ∈ H, let χ D be the quadratic Dirichlet character modulo D defined by the Kro- For more details about Dirichlet characters for polynomials over finite fields see [16,Chapters 3,4]. The L-function associated to the character χ D is defined by the following Dirichlet series From [16,Propositions 4.3,14.6 and 17.7], we have that L(s, χ D ) is a polynomial in z = q −s of degree deg(D) − 1. Also we have that the quadratic Dirichlet L-function associated to χ D , L(z, χ D ) = L(s, χ D ), has a "trivial" zero at z = 1 (resp. z = −1) if and only if deg(D) is even and sgn(D) = 1 (resp. deg(D) is even and sgn(D) = γ) and so we can define the "completed" L-function as is even and sgn(D) = 1, (1 + z) −1 L(z, χ D ) if deg(D) is even and sgn(D) = γ, which is a polynomial of even degree 2g D and satisfies the functional equation In all cases, we have that is the numerator of the zeta function associated to the hyperelliptic curve given in the affine form by C D : y 2 = D(T ).
(2.6) The following proposition is quoted from Rudnick [17], and it is proved by using the explicit formula for L(s, χ P ) and the Riemann Hypothesis for curves.  [4]). Let F q be a fixed finite field of odd cardinality with q ≡ 1 (mod 4). Then for every ε > 0 we have, and P ∈P 2g+1 The results of Andrade and Keating correspond to the average of quadratic Dirichlet Lfunction associated to the imaginary quadratic function field k( √ P ), i.e., it is the function field analogue of the Problem 1.2 with v = 3. In this paper we extend the results of Andrade and Keating by establishing the corresponding asymptotic formulas for the case of quadratic Dirichlet L-functions associated to the real quadratic function field k( √ P ), which is the function field analogue of the Problem 1.2 with v = 1. It is again worth noting that in the classical case (number fields) only asymptotics formulas for the first moment of this family are known. But in function fields, we can do better by establishing the second moment.
We also establish the corresponding asymptotic formulas for the case of quadratic Dirichlet L-functions associated to the inert imaginary quadratic function field k( √ γP ). In addition, we derive asymptotic formulas for the mean values of quadratic Dirichlet prime L-functions at s = 1 and s = 2, which are connected to the mean values of the ideal class numbers and to the cardinalities of second K-groups. Our main results are presented below. A prime L-function is the L-function associated to the quadratic character χ P where P is a prime polynomial. For the first moment of prime L-functions, we have the following theorem.
Theorem 2.3. Let F q be a fixed finite field with q being a power of an odd prime.

19)
and, for s = 1, we have (2.20) (3) For s ∈ C with Re(s) ≥ 1 2 , we have Remark 2.4. Note that in many of our estimates (e.g. when we use the O and ≪ notations) the implied constant may depend on q.
Corollary 2.5. Let q be a fixed power of an odd prime. For every ε > 0, we have (1) For the second moment of prime L-functions at s = 1 2 , we have the following theorem. Theorem 2.6. Let q be a fixed power of an odd prime. As g → ∞, we have that (1) and (2.33) Corollary 2.8. With q kept fixed power of an odd prime and g → ∞, we have (2.35) Proof. We only give the proof of (2.34). A similar argument will give the proof of (2.35). From Theorem 2.3 (2) and Theorem 2.6 (1), we have where c 1 and c 2 are the constants given in the above theorems. By applying Cauchy-Schwarz inequality we have that the number of monic irreducible polynomials P ∈ P 2g+2 such that L( 1 2 , χ P ) = 0 exceeds the ratio of the square of the quantity in (2.36) to the quantity in (2.37).
A simple computation shows that K g (1) = ζ A (2). For the mean value of prime Lfunctions at s = 1, from Theorem 2.3, we have the following: Theorem 2.9. Let q be a fixed power of an odd prime. For every ε > 0, we have (1) is even and sgn(D) = 1, is even and sgn(D) = γ. (2.41) By combining Theorem 2.9 and (2.41), we obtain the following corollary.
Corollary 2.10. Let q be a fixed power of an odd prime. For every ε > 0, we have For any non-constant square free polynomial For the mean value of prime L-functions at s = 2, as an application of Theorem 2.3, we have the following.
Theorem 2.11. Let q be a fixed power of an odd prime. For every ε > 0, we have (1) Putting together Theorem 2.11 and (2.45), we obtain the following corollary.
Corollary 2.12. Let q be a fixed power of an odd prime. For every ε > 0, we have (1) as g → ∞.
2.2. Even characteristic case. We now will handle the more difficult case. In this section we assume that q is a power of 2.
2.2.1. Quadratic function fields of even characteristic. The theory of quadratic function fields of even characteristic was first developed in [7] and we sketch below the basics on function fields of characteristic even. Every separable quadratic extension K of k is of Hence, we can normalize u to satisfy the following conditions (see [12]): . The local discriminant of K u at the infinite prime ∞ is ∞ deg(f (T ))+1 if K u is ramified imaginary and trivial otherwise. Hence, the discriminant D u of K u is given by and, by the Hurwitz genus formula, the genus g u of K u is given by In general, the normalization (2.52) is not unique. Fix an element ξ ∈ F q \ ℘(F q ). Then every u can be normalized uniquely to satisfy the following conditions: , ξ}, and α n = 0 for n > 0. Let F be the set of such u's above with n = 0 and α = 0, and F ′ be the set of such u's above with n = 0 and α = ξ. Then, we see that u → K u defines an one-to-one correspondence between F (resp. F ′ ) and the set of real (resp. inert imaginary) separable quadratic extensions of k. Similarly, if we denote by H the set of such u's above with n = 0, then u → K u defines an one-to-one correspondence between H and the set of ramified imaginary separable quadratic extensions of k.

2.2.2.
Hasse symbol and L-functions. Let P ∈ P. For u ∈ k which is P -integral, the Hasse symbol [u, P ) with values in F 2 is defined by For u ∈ k and 0 = N ∈ A, we also define the quadratic symbol: This symbol is clearly additive in its first variable, and multiplicative in the second variable. For the quadratic extension K u of k, we associate a character χ u on A + which is defined by , where ε(u) = 1 if K u is ramified imaginary and ε(u) = 0 otherwise. Also we have that L(z, χ u ) has a "trivial" zero at z = 1 (resp. z = −1) if and only if K u is real (resp. inert imaginary), so we can define the "completed" L-function as which is a polynomial of even degree 2g u satisfying the functional equation Main results. We are interested in the family of real, inert imaginary or ramified imaginary quadratic extensions K u of k whose finite discriminant is a square of prime polynomial, i.e., G u ∈ P. For any two subsets U, V of k and w ∈ k, we write Let F be the set of rational functions u ∈ F whose denominator is a monic irreducible polynomial, i.e., u = A P ∈ F with P ∈ P and 0 = A ∈ A, deg(A) < deg(P ), and F ′ = F + ξ. Then, under the above correspondence u → K u , F (resp. F ′ ) corresponds to the set of real (resp. inert imaginary) separable quadratic extensions of k whose discriminant is a square of prime polynomial. For each positive integer n, let F n be the set of rational functions u = A P ∈ F such that P ∈ P n and F ′ n = F n + ξ. Then, under the correspondence u → K u , F g+1 (resp. F ′ g+1 ) corresponds to the set of real (resp. inert imaginary) separable quadratic extensions of k whose discriminant is a square of prime polynomial and genus is g.
For each positive integer s, let G s be the set of polynomials F (T ) ∈ A 2s−1 of the form Let G be the union of G s 's for s ≥ 1 and H = F + G. Then, under the correspondence u → K u , H corresponds to the set of ramified imaginary separable quadratic extensions of k whose finite discriminant is a square of prime polynomial. For integers r, s ≥ 1, let H (r,s) = F r + G s . Then, for each u ∈ H (r,s) , the corresponding field K u is a ramified imaginary imaginary of genus r + s − 1. For integer n ≥ 1, let H n be the union of H (r,n−r) 's for 1 ≤ r ≤ n − 1. Then, under the correspondence u → K u , H g+1 corresponds to the set of ramified imaginary separable quadratic extensions of k whose finite discriminant is a square of prime polynomial and genus is g.
In this paper, we are interested in asymptotics for the sums (as q is fixed and g → ∞): For each P ∈ P, let F P be the set of rational functions u ∈ F whose denominator is P , and F ′ P := F P + ξ. Then F g+1 is disjoint union of the F P 's and F ′ g+1 is disjoint union of the F ′ P 's, where P runs over prime polynomials in P g+1 . Hence, we can write (2.68) For the first moment of such L-functions, we have the following theorem.
Theorem 2.13. Let F q be a fixed finite field with q being a power of 2. (2.70) (2) For any ε > 0 and for s ∈ C with Re(s) ≥ 1 2 and |s − 1| > ε, we have where J ′ g (s) and J * g (s) are given in Theorem 2.3 (2) and, for s = 1, we have where K ′ g (s) and K * g (s) are given in Theorem 2.3 (3).
Remark 2.14. If q is odd, the quadratic extension k( √ γD) of k is also ramified imaginary for any monic square-free polynomial D of odd degree. Under the changing of variable T → γT , k( √ γD) becomes to k( √ D). Hence, we only consider the family {k( √ P ) : P ∈ P 2g+1 } in Theorem 2.3 (1). However, if q is even, we consider all separable ramified imaginary one. This is the reason why the constant "2" appears in Theorem 2.13 (1) and does not appear in Theorem 2.3 (1).
For the second moment of L-functions at s = 1 2 , we have the following theorem. Theorem 2.15. Let q be a fixed power of 2. As g → ∞, we have There is a unique quadratic extension k( √ P ) of k whose (finite) discriminant is P in case of q being odd, but if q is even, there are Φ(P ) = |P | − 1 separable quadratic extensions K u of k whose (finite) discriminant is P 2 . For this reason, |P | 2 appears in Theorem 2. 15 whereas |P | appears in Theorem 2.6. Considering this difference, we may regard Theorem 2.15 as an even characteristic analogue of Theorem 2.6.
As in odd characteristic case, we can consider H g+1 , F g+1 and F ′ g+1 as probability spaces (ensembles) with the uniform probability measure attached to them. So the expected value of any function F on H g+1 , F g+1 or F ′ g+1 is defined as (2.81) Since (see Lemma 4.3) (2.84) From Theorems 2.13 and 2.15, we get the following corollary.
Corollary 2.17. With q kept fixed power of 2 and g → ∞, we have We follow the same reasoning as it is done in Corollary 2.8 with Theorems 2.13 and 2.15 to get the following corollary.
Corollary 2.18. With q kept fixed power of 2 and g → ∞, we have (2.92) (2.93) From Theorem 2.13, we have the following result concerning the first moment of Lfunctions at s = 1.

"Approximate" functional equations of L-functions
For any separable quadratic extension K of k, let χ K denote the character χ D if q is odd and K = k( √ D), where D is a non-constant square free polynomials D ∈ A with sgn(D) ∈ {1, γ}, or the character χ u if q is even and K = K u , where u ∈ k is normalized as in (2.57). Let L(s, χ K ) be the L-function associated to χ K . Then L(s, χ K ) is a polynomial in z = q −s of degree δ K = 2g K + 1 2 (1 + (−1) ε(K) ), where g K is the genus of K, ε(K) = 1 if K is ramified imaginary and ε(K) = 0 otherwise. Write (3.1) Lemma 3.1.
(1) For any P ∈ H g and L ∈ A + l with l ≤ g, we have χ P (L 2 ) = 1. By (4.1), we have Since A g (s) ≪ g, the error term in (4.8) is ≪ |P | 1 2 . Hence, we get the result. The proofs of (2), (3) and (4) are similar as that of (1). Now, we consider the contribution of non-squares. For any non-constant monic polynomial f , which is not perfect square, we can reformulate Proposition 2.1 as follow: (4.10) (2) For s ∈ C with Re(s) ≥ 1 2 , we have (4.12) (4) For s ∈ C with Re(s) ≥ 1 2 , we have Proof.
For P ∈ P and f ∈ A + , let Γ f,P and T f,P be defined by (see [6, §3]) Proof. By [6, Lemma 3.1], we have Γ f,P = 0, so T f,P = Γ f,P − { 0 f } = −1. Then, we have A/P f (4.57)  Suppose that P ∤ f and f is not a perfect square. Then Γ f,P,s = 0 and T f,P,s ≪ q s .

Now we consider the contribution of non-squares.
Proposition 4.9.
(1) By using the fact that g r=1 1 r ≪ log g and Lemma 4.6, we have Similarly, we can prove (2). (2) For s ∈ C with Re(s) ≥ 1 2 , we have Proof.

Proof of Theorem 2.13 (3). For any
where ν(s) := 1+q −s 1+q s−1 . Following the process as in the proof of Theorem 2.13 (2), we can show that , which is a multiplicative function on A + , and ζ ρ (s) be the Dirichlet series associated to ρ. In the proof of [4,Lemma 4.4], it is shown that Putting z = q −s and considering the power series expansion of 1−qz 2 (1−qz) 3 at z = 0, we can see that Lemma 5.3. Let P ∈ P r . Then, for any integer n ≥ 0, the value is independent of P , and depends only on r. Denote this value by ρ * n (r), and let ρ n = f ∈A + n d(f 2 ). Then we have Proof. Let ζ ρ (s) be the Dirichlet series given in Remark 5.2. Write Let ζ * ρ (s) be the power series defined by Putting z = q −s , since ρ(P n ) = d(P 2n ) = 2n + 1, we have By comparing the coefficients, we have      ρ * n = ρ n for 0 ≤ n ≤ r − 1, ρ * n + ρ * n−r = ρ n − 2ρ n−r for r ≤ n ≤ 2r − 1, ρ * n + ρ * n−r = ρ n − 2ρ n−r + ρ n−2r for 2r ≤ n.

5.2.
Odd characteristic case. In this section, we give a proof of Theorem 2.6. In §5.2.1, we obtain several results of the contribution of squares and of non-squares, which will be used to calculate the second moment of L-functions at s = 1 2 in §5.2.2 and §5.2.3. 5.2.1. Preparations for the proof. We first consider the contribution of squares.