On smooth ideals in number fields
The authors find a lower bound for \(\psi_F(x,y)\), the number of integral ideals in \(\mathbb{F}\) of norm \(\leq x\) which are \(y\)-smooth (have all prime factors of norm \(\leq y)\). The rational result, (needed for primality testing), is essentially that \(\log (\psi_\mathbb{Q} (x,y)/x)\) has order at least \(\approx \log (1/u^u)\) for large \(u= \log x/ \log y\) [see \textit{C. Pomerance}, Math. Cent. Tracts 154, 89-139 (1982; Zbl 0508.10004)]. The generalized result for an arbitrary field is much the same in magnitude, but it is bolstered with precise orders of magnitude of errors. The estimate is simplified by restriction to the case of prime ideals which split completely in the closure of \(\mathbb{F}/ \mathbb{Q}\). The GRH is invoked for the error estimates on \(\pi(w) - \text{Li}(w)/n\) with splitting primes in a normal field of degree \(n\) [\textit{J. Oesterlé}, Astérisque 61, 165-167 (1979; Zbl 0418.12005)].
lower bound for the number of smooth integral ideals
scientific article; zbMATH DE number 913831