Square-free integers as ideal norms (Q966540)

Let \(K\) be a quadratic number field, denote by \(F(n)\) be the number of ideals of \(K\) having norm \(n\), and put \(Q(x)=\sum_{n\leq x}\mu^2(n)F(n)\). The authors apply the method used by \textit{W.-G. Zhai} in the case \(K=\mathbb Q(i)\) in [Number theory. Tradition and modernization. Papers from the 3rd China-Japan seminar on number theory, Xi'an, China, 2004. New York, NY: Springer. Developments in Mathematics 15, 219--227 (2005; Zbl 1197.11129)] to show that if the error term in the Ideal Theorem for quadratic fields is \(O(x^\theta)\) with \(\theta>1/4\) (it is known that one can take \(\theta=23/73=0.315\dots\) \textit{M. N. Huxley} and \textit{N. Watt}, [Isr. J. Math. 120, Pt. A, 125--153 (2000; Zbl 0977.11049)], then for \(y\gg x^\alpha\) with \(\alpha>\theta\) the difference \(Q(x+y)-Q(x)\) is asymptotic to \(Ay\), where \(A\) denotes the residue of the Dedekind zeta-function of \(K\) at \(s=1\).