Improving the Error Term in the Mean Value of in the Hyperelliptic Ensemble

Abstract

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

Hybrid Euler-Hadamard product for quadratic Dirichlet L-functions in function fields

“…The first three lemmas are in [, Lemma 2.2, Proposition 3.1 and Lemma 3.2]. Lemma For we have where the summations over are over monic polynomials whose prime factors are among the prime factors of .…”

“…They explicitly computed the main term, which is of size , and bounded the error term by . This result was recently improved by Florea with a secondary main term and an error term of size . Florea's approach is similar to Young's , but in the function field setting, it is striking that one can surpass the square‐root cancellation.…”

Mean value of the class number in function fields revisited

“…Recently, Florea [7] established an asymptotic formula for the first moment of quadratic Dirichlet Lfunctions in function fields. She obtained a strenuous lower order term that had not appeared in the previous work of Hoffstein and Rosen [11] and Andrade and Keating [3].…”

“…In this paper we use the Poisson summation, as developed by Florea in [7], to obtain an extra main term for the average value of the class number in function fields. With this, we are able to go beyond the results of Hoffstein and Rosen [11], Andrade [1] and Jung [12] related to the average value of the class number.…”

Mean values of derivatives of L-functions in function fields: I

“…first moment of quadratic L-functions over function fields, were firstly studied by Hoffstein and Rosen [10] and recently by Andrade and Keating [3], Florea [7] and Jung [12].…”