On the distribution of irreducible algebraic integers (Q1006430)

Let \(K\) be an algebraic number field with class-number \(h\geq2\), let \(M_K\) be the set of all principal ideals of \(K\), generated by irreducible elements, let \[ \zeta(s,M_K)=\sum_{I\in M_K}{1\over N(I)^s} \] be the corresponding zeta-function and put \[ T_K(x)={1\over2\pi i}\int_{\Gamma}\zeta(s,M_K){x^s\over s}ds, \] where \(\Gamma\) is a suitable curve surrounding the pole of \(\zeta(s,M_K)\) at \(s=1\). It follows from earlier results of the author [Acta Arith. 43, 53--68 (1983; Zbl 0526.12006)] that for the counting function \(M_K(x)\) of \(M_K\) one has \[ M_K(x)=T_K(x)+E_K(x) \] with \[ E_K(x)=O\left(x\exp\left(-a_K\sqrt{\log x}\right)\right). \] The author shows under certain assumptions about the multiplicity of zeros of Hecke \(L\)-functions corresponding to \(K\) that the error term \(E_K(x)\) has oscillations of logarithmic frequency and size \[ \sqrt x(\log\log x)^{D(K)-1}/\log x, \] \(D(K)\) denoting the constant of Davenport of the class-group of \(K\). He shows also that the assumption about \(L\)-functions is satisfied if either \(h=2\) or \(K\) is of prime degree \(p\) and the Dedekind zeta function of the Hilbert class-field of \(K\) has at least one non-trivial zero of multiplicity not divisible by \(p\).