Density of cubic field discriminants (Q2723537)

The author deals with the approximations to \(g_\pm(x)\), the number of cubic fields \(k\) of \(|\text{disc}(k)|\leq x\), according to the \(\pm\)-sign of \(\text{disc} (k)\). The abelian subcase was investigated by the reviewer [\textit{H. Cohn}, Proc. Am. Math. Soc. 5, 476-477 (1954; Zbl 0055.26901)], and although the corresponding number \(f(x)\) is \(\approx\sqrt x\), the value of \(f/3\) is added to \(g_+\) for ``error smoothness'' to make \(h_+\), while \(h_-=g_-\). Then the Davenport-Heilbronn estimate is \(h_\pm \approx C_\pm x/(12\zeta (3))\), with \(C_+=1\), \(C_-=3\) [\textit{H. Davenport} and \textit{H. Heilbronn}, Proc. R. Soc. Lond., Ser. A 322, 405-420 (1971; Zbl 0212.08101)]. The problem is that the error is relatively large, as noted by \textit{G. W. Fung} and \textit{H. C. Williams} [Math. Comput. 55, 313-325 (1990; Zbl 0705.11063)].NEWLINENEWLINENEWLINEThe author mainly gives a heuristic argument that the error is \(c_\pm x^{5\over 6}\) with \(K_+=1\), \(K_-=\sqrt 3\), and NEWLINE\[NEWLINEc_\pm=K_\pm 4\zeta (1/3)/ \bigl(5\Gamma(2/3)^3 \zeta(5/3) \bigr)NEWLINE\]NEWLINE (which is negative because of \(\zeta (1/3))\). The exponent 5/6 comes from the pole of a Dirichlet series used by \textit{T. Shintani} [J. Math. Soc. Japan 24, 132-188 (1972; Zbl 0227.10031)]. Also see \textit{B. Datskovsky} and \textit{D. J. Wright} [J. Reine Angew. Math. 386, 116-138 (1988; Zbl 0632.12007)]. Further error is believed to be \(o(\sqrt x)\), from computations.