Moments of central values of cubic Hecke \(L\)-functions of \(\mathbb{Q}(i)\) (Q2297932)

The authors consider \(L\)-functions of cubic characters in two fields: the field \(K={\mathbb Q}(i)\) of Gaussian integers and the field \(F={\mathbb Q}(\zeta_{12})={\mathbb Q}(e(1/12))\). They estimate (Theorem 1.1) a sum of central values of such \(L\)-functions in \(K\) weighted with a smooth, compactly-supported function \(w\): \[ \sum_{q\in{\mathcal O}_K/{\mathcal O}_K^\times,(q,6)\sim 1} \sum_{\text {primitive cubic }\chi\text{ mod }q} L(1/2,\chi)w(N_K(q)/Q) = C_0 Q \hat w(0) + O(Q^{37/38+\varepsilon}). \] They also give (Theorem 1.2) an upper bound for the second moment: \[ \sum_{q\in{\mathcal O}_K/{\mathcal O}_K^\times,(q,6)\sim 1} \sum_{\text {primitive cubic }\chi\text{ mod }q} \lvert L(1/2,\chi)\rvert^2 = O(Q^{11/9+\varepsilon}(1+\lvert t \rvert)^{1+\varepsilon}). \] For characters of \(F\) they give a similar upper bound for the sum of squares, however, the cubic characters involved are of a special type, parametrized by elements of \({\mathcal O}_K\). The authors also mention a lower bound on the number of primitive cubic characters \(\chi\) of \(K\) of a given norm, satisfying \(L(1/2,\chi)\neq 0\), as consequence of these estimates.