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

Abstract

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

$$\sum\limits_{n \leqslant x} {F(n)F(n + r) = M_K (r)x + E_K (x,r).} $$

HereM _k (r) is an explicitly determined function ofr which depends onK, and for every ε>0 the error term is bounded by$|E_K (x,r)|<< |D|^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2} + \varepsilon } x^{{5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-\nulldelimiterspace} 6} + \varepsilon }$ uniformly for$r<< |D|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} x^{{5 \mathord{\left/ {\vphantom {5 6}} \right. \kern-\nulldelimiterspace} 6}}$ Moreover,E _k (x,r) is small on average, i.e$\int_X^{2X} {|E_K (x,r)|^2 dx}<< |D|^{4 + \varepsilon } X^{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2} + \varepsilon }$ uniformly for$r<< |D|X^{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-\nulldelimiterspace} 4}}$.