ISSN:
1436-5081
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 $$r〈〈 |D|X^{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-\nulldelimiterspace} 4}} $$ .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01501132
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