Improving the error term in the mean value of L(12,χ) in the hyperelliptic ensemble

الأصل · EN

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, square-free polynomials D of degree 2g+1, the main term is of size |D| |D| (where |D|=q²ᵍ⁺¹) and Andrade and Keating bound the error term by |D| 34+ (2)/2. For simplicity, we assume that q is prime with q ≡ 1 4. We prove that there is an extra term of size |D|¹/³ |D| in the asymptotic formula and bound the error term by |D|¹/⁴⁺ε.