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

LetK be a quadratic number field with discriminantD and denote byF(n) the number of integral ideals with norm equal ton. Forr≥1 the following formula is proved $$sumlimits_{n leqslant x} {F(n)F(n + r) = M_K (r)x + E_K (x,r).} $$ HereM _k (r) is an explicitly determined function ofr which depends onK, and for every ε>0 the error term is bounded by (|E_K (x,r)|<< |D|^{{3 mathord{left/ {vphantom {3 2}} right. kern-0em} 2} + varepsilon } x^{{5 mathord{left/ {vphantom {5 6}} right. kern-0em} 6} + varepsilon } ) uniformly for (r<< |D|^{{1 mathord{left/ {vphantom {1 2}} right. kern-0em} 2}} x^{{5 mathord{left/ {vphantom {5 6}} right. kern-0em} 6}} ) Moreover,E _k (x,r) is small on average, i.e (int_X^{2X} {|E_K (x,r)|^2 dx}<< |D|^{4 + varepsilon } X^{{5 mathord{left/ {vphantom {5 2}} right. kern-0em} 2} + varepsilon } ) uniformly for (r<< |D|X^{{3 mathord{left/ {vphantom {3 4}} right. kern-0em} 4}} ) .