About one analog of the additive divisor problem with quadratic forms

L. N. Kurtova

Čebyševskij sbornik, Tome 15 (2014) no. 2, pp. 33-49.

Résumé

In the number theory additive problems is very important. One of them is the Ingam binary additive divisor problem on the representation of natural number as the difference of product of numbers. Many mathematician like T. Esterman, D. I. Ismoilov, D. R. Heath-Brown, G. I. Arkhipov and V. N. Chubarikov, J.-M. Deshouillers and H. Iwaniec improved the remainder term in the asymptotic formula of the number of solution of this diophantine equation. In present paper one problem with quadratic forms is considered. This problem is analog of the Ingam binary additive divisor problem. Let $d$ — negative square-free number, $F={Q}(\sqrt{d})$ — imaginary quadratic field, $\delta_{F}$ — discriminant of field $F$, $Q_{1}(\overline{m})$, $Q_{2}(\overline{k})$ — binary positive defined primitive quadratic forms with matrixes $A_{1}$, $A_{2}$, $\det A_{1}=\det A_{2}=-\delta_{F}$, $\varepsilon>0$ — arbitrarily small number; $n\in \mathbb{N}$, $h \in \mathbb{N}$. The asymptotical formula of the number of solution of diophantine equation $Q_{1}(\overline{m})-Q_{2}(\overline{k})=h$ with weight coefficient $\exp\left({-({Q_{1}(\overline{m})+Q_{2}(\overline{k})})/{n}}\right)$ is received. In this asymptotical formula discriminant of field $\delta_F$ is fixed and the remainder term is estimating as $O(h^{\varepsilon}n^{3/4+\varepsilon})$, which not depend of $\delta_F$. Moreover the parameter $h$ grow as $O(n)$ with growing on the main parameter $n$. Proof of the asymptotical formula based on circular method when sum, which is solution of diophantine equation, may be representing as integral. Interval of integration divided by numbers of Farey series. The taking weight coefficient allow to use the functional equation of the theta-function. Moreover the estimation of one sum with Gauss sums is important. Using the evident formula of some product of Gauss sums of the number which coprimes of discriminant of field this sum represented of Kloosterman's sum which estimate by A. Weil. Bibliography: 16 titles.

Détail

Keywords: additive problems of number theory, asymptotic formula, Kloosterman's sum, quadratic form.

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