On the representation of positive integers by quadratic forms with seven variables (Q1769439)

The authors consider the numbers \(r(n;Q_k)\) of representations of a positive integer \(n\) by the positive definite quadratic forms \(Q_k = 2\sum_{i=1}^kx_i^2+\sum_{j=k+1}^7x_j^2\) in seven variables, where \(1 \leq k \leq 6\). Formulas of the type \(r(n;Q_k) = \rho(n;Q_k) + \vartheta(n;Q_k)\) are obtained for these numbers, where \(\rho(n;Q_k)\) is the singular series and \(\vartheta(n;Q_k)\) is a Fourier coefficient of a cusp form.