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On the Representation of Numbers by the Direct Sums of Some Binary Quadratic Forms

Authors:	Kachakhidze N

Abstract:	The systems of bases are constructed for the spaces of cusp forms $S_k (\Gamma _0 (3),\chi )(k \geqslant 6),S_k (\Gamma _0 (7),\chi )(k \geqslant 3)$ and $S_k (\Gamma _0 (11),\chi )(k \geqslant 3)$ . Formulas are obtained for the number of representations of a positive integer by the sum of k binary quadratic forms of the kind $x_1^2 + x_1 x_2 + x_2^2 (6 \leqslant k \leqslant 17)$ , of the kind $x_1^2 + x_1 x_2 + 2x_2^2 (3 \leqslant k \leqslant 11)$ and of the kind $x_1^2 + x_1 x_2 + 3x_1^2 (3 \leqslant k \leqslant 7)$ .

Keywords:	Representation of numbers by quadratic forms direct sum of quadratic forms space of cusp forms generalized multiple theta-series
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