Andrade and Keating computed the mean value of quadratic Dirichlet L-functions at the critical point, in the hyperelliptic ensemble over a fixed finite field Fq. Summing L(1/2, χD) over monic, squarefree polynomials D of degree 2g +1, the main term is of size |D| log q |D| (where |D| = q 2g+1 ) and Andrade and Keating bound the error term by |D| 3 4 + log q (2) 2. For simplicity, we assume that q is prime with q ≡ 1 (mod 4). We prove that there is an extra term of size |D| 1/3 log q |D| in the asymptotic formula and bound the error term by |D| 1/4+ǫ .Goldfeld and Hoffstein [8] improved the error bound to D 19/32+ǫ . Young [22] considered the smoothed first moment and showed that the error term is bounded by D 1/2+ǫ . The remainder term for the first moment of quadratic Dirichlet L-functions is conjectured to be of size D 1/4+ǫ in [1]. Our approach in bounding the remainder over function fields is similar to Young's method in [22], but in our setting, we are able to go beyond the square-root cancellation.Jutila [11] also computed the variance, and Soundararajan [20] computed the second and third moments, when averaging over real, primitive, even characters with conductor 8d. It is conjectured thatwhere the sum is over fundamental discriminants. Keating and Snaith [14] conjectured a precise value for C k , using analogies with random matrix theory. There is another conjecture of Conrey, Farmer, Keating, Rubinstein and Snaith [5] for the integral moments, and the formulas include all the principal lower order terms. The conjecture agrees with the computed first three moments.In the function field setting, the analogous problem is to find asymptotics for( 1.2) as |D| = q deg(D) → ∞, where H 2g+1,q denotes the space of monic, square-free polynomials of degree 2g + 1Since we let |D| → ∞, we can consider two limits: the limit q → ∞ (and g fixed), or g → ∞ (and q fixed). Katz and Sarnak [12], [13] used equidistribution results to relate the q-limit of (1.2) to a random matrix theory integral, which was then computed by Keating and Snaith [14].Here, we are interested in the other limit, when g → ∞ and q is fixed. In analogy with the conjectured moments for the family of quadratic Dirichlet L-functions over number fields, Andrade and Keating [4] conjectured asymptotic formulas for integral moments of L(1/2, χ D ), for q fixed and g → ∞.In the recent paper [18], Rubinstein and Wu provide numerical evidence for the conjecture in [4]. They numerically computed the moments for k ≤ 10, d ≤ 18 (where d = 2g + 1) and various values of q and compared them to the conjectured formulas. Their data suggest that the ratio of the actual moment to the conjectured moment goes to 1 as g grows.Note that we can also compute the shifted first moment D∈H2g+1 L 1 2 + α, χ D , when α is in a small neighborhood of 0. Working instead with the completed L-function Λ(s, χ D ) = q −g(1−2s) L(s, χ D ) (which

Abstract. We obtain an asymptotic formula for the fourth moment of quadratic Dirichlet Lfunctions over Fq [x], as the base field Fq is fixed and the genus of the family goes to infinity. According to conjectures of Andrade and Keating, we expect the fourth moment to be asymptotic to q 2g+1 P (2g + 1) up to an error of size o(q 2g+1 ), where P is a polynomial of degree 10 with explicit coefficients. We prove an asymptotic formula with the leading three terms, which agrees with the conjectured result.

We study the 1-level density and the pair correlation of zeros of quadratic Dirichlet L-functions in function fields, as we average over the ensemble H 2g+1 of monic, square-free polynomials with coefficients in F q [x]. In the case of the 1-level density, when the Fourier transform of the test function is supported in the restricted interval ( 1 3 , 1), we compute a secondary term of size q − 4g 3 /g, which is not predicted by the Ratios Conjecture. Moreover, when the support is even more restricted, we obtain several lower order terms. For example, if the Fourier transform is supported in ( 1 3 , 1 2 ), we identify another lower order term of size q − 8g 5 /g. We also compute the pair correlation, and as for the 1-level density, we detect lower order terms under certain restrictions; for example, we see a term of size q −g /g 2 when the Fourier transform is supported in ( 1 4 , 1 2 ). The 1-level density and the pair correlation allow us to obtain non-vanishing results for L( 1 2 , χ D ), as well as lower bounds for the proportion of simple zeros of this family of L-functions.

We obtain asymptotic formulas for the second and third moment of quadratic Dirichlet L-functions at the critical point, in the function field setting. We fix the ground field Fq, and assume for simplicity that q is a prime with q ≡ 1 (mod 4). We compute the second and third moment of L(1/2, χ D ) when D is a monic, square-free polynomial of degree 2g + 1, as g → ∞. The answer we get for the second moment agrees with Andrade and Keating's conjectured formula in [3]. For the third moment, we check that the leading term agrees with the conjecture.

We develop a hybrid Euler‐Hadamard product model for quadratic Dirichlet L–functions over function fields (following the model introduced by Gonek, Hughes and Keating for the Riemann‐zeta function). After computing the first three twisted moments in this family of L–functions, we provide further evidence for the conjectural asymptotic formulas for the moments of the family.