On the distribution of ideals in cubic number fields

LetK be a cubic number field. Denote byA _K (x) the number of ideals with ideal norm ≤x, and byQ _K (x) the corresponding number of squarefree ideals. The following asymptotics are proved. For every ε>0 ε>0 $$begin{gathered} {text{ }}A_K (x) = c_1 x + O(x^{43/96 + in } ), hfill Q_K (x) = c_2 x + O(x^{1/2} exp {text{ }}{ - c(log {text{ }}x)^{3/5} (log log {text{ }}x)^{ - 1/5} } ). hfill end{gathered}$$ Herec ₁,c ₂ andc are positive constants. Assuming the Riemann hypotheses for the Dedekind zeta function ζ _K , the error term in the second result can be improved toO(x ^53/116+ε).