Minimal generators of the ideal class group (Q1998892)

Let \(K\) be an algebraic number field and denote by \(d_K\) the absolute value of the discriminant of \(K\). Let \(T(K)\) be the minimal value of \(T\) such that the set of degree one prime ideals of norm \(\le T\) generates the class-group of \(K\). It has been shown by \textit{E. Bach} [Math. Comput. 55, No. 191, 355--380 (1990; Zbl 0701.11075)] that \(GRH\) implies \(T(K)\le12\log^2(d_K)\) and later \textit{L. Grenié} and \textit{G. Molteni} [Math. Comput. 87, No. 313, 2483--2511 (2018; Zbl 1461.11137)] replaced \(12\) by \(4.01\). The best known unconditional bound \[ T(K)\le c\sqrt{d_K} \] with \(c=(50.7)^{-r_1/2}(19.9)^{-r_2}\) has been given by \textit{R. Zimmert} [Invent. Math. 62, 367--380 (1981; Zbl 0456.12003)]. The author shows that for certain families of quadratic, non-normal cubic and cyclic quartic and quintic fields one can unconditionally establish the bound of the form \(T(K)=O(\log^\alpha(d_K))\) for almost all fields in the family. This is achieved by showing that such bound holds if one assumes the non-vanishing in the region \([1-2/\alpha,1]\times[-2\log^3(d_K),2\log^3(d_K)]\) (for some \(\alpha>4\)) of \(\zeta_K(s)/\zeta(s)\) and a class-group \(L\)-function attached to \(K\), and applies the results of \textit{E. Kowalski} and \textit{P. Michel} [Pac. J. Math. 207, No. 2, 411--431 (2002; Zbl 1129.11316)] on zeros of automorphic \(L\)-functions.