### 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 $n\geq 2$.