The second and third moment of L(1/2,χ) in the hyperelliptic ensemble

Abstract 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 𝔽 q {\mathbb{F}_{q}} , and assume for simplicity that q is a prime with q ≡ 1 ⁢ ( mod ⁢ 4 ) {q\equiv 1~{}(\mathrm{mod}\,4)} . We compute the second and third moment of L ⁢ ( 1 / 2 , χ D ) {L(1/2,\chi_{D})} , when D is a monic square-free polynomial of degree 2 ⁢ g + 1 {2g+1} , as g → ∞ {g\to\infty} . The answer we get for the second moment agrees with Andrade and Keating’s conjectured formula in [4]. For the third moment, we check that the leading term agrees with the conjecture.

摘要：在函数域情形下，我们得到了二次狄利克雷\(L\)-函数在临界点的二阶矩和三阶矩的渐近公式。我们固定基域\(\mathbb{F}_{q}\)，并且为简便起见，假设\(q\)是一个满足\(q\equiv 1\ (\mathrm{mod}\ 4)\)的素数。当\(D\)是一个次数为\(2g + 1\)的首一、无平方因子多项式且\(g\rightarrow\infty\)时，我们计算\(L(1/2,\chi_{D})\)的二阶矩和三阶矩。我们得到的二阶矩的结果与安德拉德和基廷在[4]中推测的公式一致。对于三阶矩，我们验证了其主项与推测相符。