Gao, Peng; Zhao, Liangyi Moments of central values of cubic Hecke \(L\)-functions of \(\mathbb{Q}(i)\). (English) Zbl 1487.11080 Res. Number Theory 6, No. 1, Paper No. 14, 22 p. (2020). 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. Reviewer: Maciej Radziejewski (Poznań) MSC: 11L40 Estimates on character sums 11R16 Cubic and quartic extensions 11R42 Zeta functions and \(L\)-functions of number fields Keywords:cubic Hecke character; cubic large sieve; moments of Hecke \(L\)-functions PDFBibTeX XMLCite \textit{P. Gao} and \textit{L. Zhao}, Res. Number Theory 6, No. 1, Paper No. 14, 22 p. (2020; Zbl 1487.11080) Full Text: DOI arXiv References: [1] Baier, S.; Young, MP, Mean values with cubic characters, J. Number Theory, 130, 4, 879-903 (2010) · Zbl 1204.11135 [2] Blomer, V.; Goldmakher, L.; Louvel, B., \(L\)-functions with \(n\)-th order twists, Int. Math. Res. Not., 2014, 7, 1925-1955 (2014) · Zbl 1328.11090 [3] Brubaker, B.; Friedberg, S.; Hoffstein, J., Cubic twists of GL(2) automorphic \(L\)-functions, Invent. 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