On zeros of cubic \(L\)-functions
Let \(\omega= (-1+\sqrt{-3})/2\) and let \(c\equiv 1\pmod 9\) be a square-free integer in \(\mathbb{Z}[\omega]\), with \(c\neq 1\). The \(L\)-functions considered in this paper are \(L(s,\chi_c)\), where \(\chi_c= (\cdot/c)_3\) is the cubic residue symbol. The first result is the zero-density estimate \[ \sum_{N(c)\leq Q} N(\sigma,T,\chi_c)\ll_\varepsilon Q^{10(1-\sigma)} T(QT)^\varepsilon, \] for \(T\geq 2\) and any \(\varepsilon> 0\), where \(c\) is restricted as above, and \(N(\sigma,T,\chi_c)\) counts zeros of \(L(s,\chi_c)\). It seems to the reviewer that one should be able to get better results for the larger sum \[ \sum_{N(c)\leq Q}\;\sum_{\chi(\operatorname{mod} c)} N(\sigma,T,\chi), \] where \(\chi\) runs over all primitive characters to modulus \(c\). Indeed a very special case of the work of \textit{J. Hinz} [Acta Arith. 31, 167--193 (1976; Zbl 0304.12011)] already gives an upper bound \(O((QT)^{4(1-\sigma)}(QT)^\varepsilon)\). The second main result is that for each fixed \(m\in\mathbb{N}\) there is an explicit positive constant \(m\) such that \[ \sum_c|L(1, \chi_c)|^{2m} \exp(- N(c)/T)\sim C_m T\quad (T\to \infty), \] the sum being over all \(c\) satisfying the restrictions above. The proofs use the reviewer's mean value theorem for cubic character sums [Isr. J. Math. 120, Pt. A, 97--124 (2000; Zbl 0989.11042)].
scientific article; zbMATH DE number 5151593