On ideals free of large prime ideal factors

Eira J. Scourfield
Royal Holloway, University of London

In J. Number Theory 32 (1989), pp 78 - 99, E.Saias established an asymptotic formula for the number $\Psi (x,y)$ of positive integers not exceeding $x$ with no prime factor exceeding $y$. His result had a very good error term and was valid in the region

H_{\epsilon }: (\log \log x)^{(5/3)+\epsilon }\leq \log y\leq \log x, for x\geq x_{0}(\epsilon )

for arbitrary $\epsilon >0$. We consider the analogous problem for an algebraic number field $K$ of degree $n\geq 2$. In the ring of integers of $K$, let $\Psi _{K}(x,y)$ denote the number of ideals with norm $\leq x$ and with no prime ideal factor with norm $>y$. We describe a result that yields an asymptotic formula for $\Psi _{K}(x,y)$ valid in $H_{\epsilon }$ and with an error term of the same order of magnitude as that in Saias's result. If $\lambda _{K}$ denotes the residue of the Dedekind zeta-function for the field $K$ at $s=1$, we have further that $\Psi _{K}(x,y)-\lambda _{K}\Psi (x,y)$ has the same order of magnitude as the second term in $\Psi (x,y)$ provided that a certain constant depending on $K$ does not vanish. The proofs of these theorems are analytic and use deep properties of the Dedekind zeta-function. We illustrate our results with an application analogous to that of estimating the sum $\sum\limits_{n\leq x}(P(n))^{-1}$ in $\mathbf{Q}$, where $P(1)=1$ and $P(n)$ is the greatest prime factor of $n\geq 2$.