A number of solutions the equation with quadratic forms of different discriminants

Author(s):  L.N. Kurtova, candidate of Sciences, no, Belgorod National Research University, Belgorod, Russia, kurtova@bsu.edu.ru

Issue:  Volume 51, № 3

Rubric:  Physics. Mathematical modeling

Annotation:  In this article, the problem with quadratic forms is considered. This problem is analog of the Ingam binary additive divisor problem. The asymptotical formula of the number of solution of diophantine equation Q1(m) — Q2(k) = h is received. Binary positive defined primitive quadratic forms Q1(m) and Q2 (k) corresponded to the ideal classes of two imaginary quadratic fields of different fixed discriminants. The number of solutions searched with weights exp(—(Q1 (m) + Q2(k))/n) with the growth of the parameter n. Proof of the asymptotical formula based on circular method. Using the evident formula of Gauss sums of the number, which coprimes of discriminants of fields, this sum represented of Kloosterman’s sum which estimate by A. Weil.

Keywords:  additive problem, asymptotic formula, number of solutions, double Gauss sum, Kloos- terrnan sum

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