On the Ono invariants of imaginary quadratic number fields (Q838441)

Let \(D\) be the discriminant of a complex quadratic number field \(K\), and let \(\nu(n)\) denote the number of prime factors (counted with multiplicity) of an integer \(n\). The Ono number \(p(D)\) is defined as the maximum of the values \(\nu(f_D(x))\) for integers \(0 \leq x \leq \frac14 D-1\), where \(f_D(x) = x^2 - D/4\) if \(4 \mid D\), and \(f_D(X) = x^2 + x - (D-1)/4\) if \(D \equiv 1 \bmod 4\). In this article, the author shows that there are only finitely many \(D < 0\) with \(p(D) = h(D)\), the class number of \(K\) (this result was proved by \textit{J. Cohen} and \textit{J. Sonn} [J. Number Theory 95, No. 2, 259--267 (2002; Zbl 1082.11068)], as well as by \textit{F. Sairaji} and \textit{K. Shimizu} [Proc. Japan Acad., Ser. A 78, No. 7, 105--108 (2002; Zbl 1052.11070]). All discriminants \(> -2 \cdot 10^8\) with this property are listed, and it is proved that there is at most one such discriminant \(D < -2 \cdot 10^8\), whose existence would contradict a special case of the Generalized Riemann Hypothesis.