On the distribution of ideals in cubic number fields (Q1118644)

Let K be a cubic number field. Denote by \(A_ k(x)\) the number of integral ideals with norm \(\leq x\), and by \(Q_ K(x)\) the corresponding number of squarefree ideals. Landau's classical asymptotical result \(A_ K(x)=\rho_ Kx+O(x^{1/2})\) can be used to prove \(Q_ K(x)=cx+O(x^{1/2})\). Here \(\rho_ K\) denotes the residue of the Dedekind zeta function \(\zeta_ K(s)\) at \(s=1\) and \(c=\rho_ K/\zeta_ K(2)\). Both estimates can be refined. For every \(\epsilon >0\) the result reads \[ A_ K(x)=\rho_ Kx+O(x^{43/96+\epsilon}), \] \[ Q_ K(x)=cx+O(x^{1/2}\quad \exp (-c(\log x)^{3/5}\quad (\log \log x)^{- 1/5}), \] where c denotes a positive constant. Assuming the Riemann hypothesis for the Dedekind zeta-function \(\zeta_ k\) the error term in the second result can be improved to \(O(x^{53/116+\epsilon}).\)