On the dimension of some spaces of generalized theta-series (Q619304)

\textit{F. Gooding} [J. Number Theory 9, 36--47 (1977; Zbl 0349.10016)] calculated exactly the dimension of the space of generalized theta-series \(T(n,Q)\) for reduced binary quadratic forms. Moreover, Gooding [Modular forms arising from spherical polynomials and positive definite quadratic forms, PhD thesis, University of Wisconsin (1971)] obtained the upper bound \(\dim T(n,Q)\leq\binom{{n\over 2}+m-2}{m-2}\) for diagonal quadratic forms. Later K. Shavgulidze obtained an upper bound for the dimension of the space \(T(n,Q)\) for some ternary and quaternary quadratic forms and calculated the dimension exactly in some cases by constructing the bases of these spaces. The author [Šiauliai Math. Semin. 3(11), 79--84 (2008; Zbl 1220.11053)] improved the upper bound for the dimension of the space \(T(4,Q)\) for diagonal quadratic forms with two equal coefficients. In this paper, the author improves the upper bound for the dimension of the space \(T(6,Q)\) for some diagonal quadratic forms of any number of variables \(b_ix_i^2\). Theorem 1. (1) Let \(b_1 = b_2\). Then \[ \dim T(6,Q) \leq \binom{m+1}{3}-\frac{m^2-3m+4}{2}. \] (2) Let \(b_1 = b_2 = \dots = b_k\) \((k\geq 3)\). Then \[ \dim T(6,Q)\leq \frac 16(m-k)^3 +\frac 12(m-k)^2 +\frac 43(m-k)+1. \]