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

The systems of bases are constructed for the spaces of cusp forms Sk(Γ0(3), χ) (k≥6), Sk(Γ0(7), χ) (k≥3) and Sk(Γ0(11), χ) (k ≥ 3). Formulas are obtained for the number of representation of a positive integer by the sum of k binary quadratic forms of the kind x1 +x1x2 +x 2 2 (6 ≤ k ≤ 17), of the kind x 2 1 +x1x2 +2x 2 2 (3 ≤ k ≤ 11) and of the kind x1 + x1x2 + 3x 2 1 (3 ≤ k ≤ 7). Let Fk denote a direct sum of k binary quadratic forms F1 with the same negative discriminant −q (q is prime ≡ 3 (mod 4)). The question of the representation of natural numbers by the direct sum of some binary quadratic forms was for the first time considered by Petersson [1]. In particular, he constructed the basis for the space of cusp forms Sk(Γ0(3), χ) for an arbitrary integer k ≥ 6 using the Dedekind η-function and obtained formulas for the number of representation r(n, Fk) of the positive integer n by Fk when F1 = x1 + x1x2 + x 2 2 and 2 ≤ k ≤ 6. Lomadze [2, 3] obtained formulas for r(n, Fk) when F1 = x1 + x1x2 + x 2 2 (2 ≤ k ≤ 17) and F1 = x1 + x1x2 + 2x2 (2 ≤ k ≤ 11) and showed that they obey a fairly definite law. To this end, he constructed the bases of the spaces Sk(Γ0(3), χ) and Sk(Γ(7), χ) apart for each k using a generalized multiple theta series. Merzlyakov [4] constructed the basis of the space Sk(Γ0(3), χ) for an arbitrary integer k ≥ 6 and obtained formulas for r(n, Fk) when F1 = x1 + x1x2 + x 2 2 (2 ≤ k ≤ 11). But these formulas do not obey any law. In the present paper, using the generalized multiple theta series, the systems of bases for the space of cusp forms Sk(Γ0(3), χ) with an integer k ≥ 6 and for the spaces Sk(Γ0(7), χ) and Sk(Γ0(11), χ) with an integer k ≥ 3 are constructed. After that, these bases are used to obtain the formulas for r(n, Fk) when F1 = x1 + x1x2 + x 2 2 (6 ≤ k ≤ 17), F1 = x1 + x1x2 + 2x2 1991 Mathematics Subject Classification. 11F11, 11F13, 11F27, 11E25.