The Period-Three Secondary Term in the Mean Value of $L(1/2,\chi_D)$ in the Hyperelliptic Ensemble

Abstract

Let $q$ be an odd prime power. We revisit Florea's asymptotic formula for the first moment of quadratic Dirichlet $L$-functions over the odd-degree hyperelliptic ensemble $\mathcal H_{2g+1}$, and show that its secondary term is not the complete one. The square-dual generating function has three double poles of the same modulus on the secondary circle: the positive real one reproduces Florea's polynomial, while the two nonreal conjugate poles contribute at the same order. The complete secondary term is $q^{2g/3}(\alpha_{g\bmod3}g+\beta_{g\bmod3})$, with real coefficients whose dependence on the genus has minimal period exactly three, and on at least two of the three residue classes the nonreal poles contribute a term of order $gq^{2g/3}$. We also prove the formula for every odd prime power $q$, whereas Florea assumes $q\equiv1\bmod4$ throughout; this is achieved by a phase-normalized quadratic Gauss sum that restores the multiplicativity and the local evaluations her argument uses. Exact finite-ensemble moments for $q=3$ exhibit the three residue-class trends.