A secondary term in the first moment of cubic $L$-functions over function fields

Abstract

Let $q$ be a fixed odd prime power with $q\equiv 2(\mathrm{mod}3)$. We study the first moment of primitive cubic Dirichlet $L$-functions over $\Fq(T)$, ordered by their genus. By averaging the rational sieve variable in the double Dirichlet series of Hong and Fang, we obtain meromorphic continuation beyond the secondary polar divisor. Combining this continuation with the cubic Gauss-sum residues computed by David, Florea and Lalín gives an explicit term of order $q^{5g/6}$, with a real coefficient depending on $g$ modulo $6$, and an error $O_{q,\eps}(q^{(23/28+\eps)g})$. The periodic coefficient is not identically zero. The proof uses an exact first difference supplied by the functional equation, an Euler product for the sieve average, and Bochner's tube theorem in the Dirichlet variables.