On the asymptotic behaviour of the ideal counting function in quadratic number fields (Q749602)

Let \(K\) be a quadratic number field with discriminant \(D\) and denote by \(F(n)\) the number of ideals with norm equal to \(n\). Then \(F(n)=F_{\chi_D}(n):=\sum_{d\mid n}\chi_D(d)\), where \(\chi_D\) is a real primitive Dirichlet character modulo \(| D|\). More generally, consider arbitrary primitive Dirichlet characters \(\chi\) and \(\Psi\) modulo \(k>1\) and the corresponding functions \(F_{\chi}(n)\) and \(F_{\Psi}(n)\). For \(r\geq 1\) the following asymptotic formula is proved \[ \sum_{n\leq x}F_{\chi}(n)F_{\Psi}(n+r) = M_{\chi,\Psi}(r)x + E_{\chi,\Psi}(x,r). \] Here \(M_{\chi,\Psi}(r)\) is an explicitly determined function of \(r\) which depends on \(\chi\) and \(\Psi\), and for every \(\varepsilon >0\) the error term is bounded by \[ | E_{\chi,\Psi}(x,r)| \ll k^{3/2+\varepsilon}X^{5/6+\varepsilon}\] uniformly for \(r \ll k^{1/2}x^{5/6}\). Moreover, \(E_{\chi,\Psi}(x,r)\) is small on average, i.e. \[ \int^{2X}_{X}| E_{\chi,\Psi}(x,r)|^2\,dx \ll k^{4+\varepsilon}X^{5/2+\varepsilon} \] uniformly for \(r \ll kX^{3/4}\).