Evaluation of the convolution sum sigma(al+bm=n) sigma(l)sigma(m) for (a, b) = (1, 48), (3, 16), (1, 54), (2, 27)

Abstract

We determine the convolution sum Sigma(al+bm=n) sigma(l)sigma(m) for (a, b) = (1, 48), (3, 16), (1, 54), (2, 27) for all positive integers n. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of n by the octonary quadratic forms k(x(1)(2) +x(1)x(2) + x(2)(2) + x(3)(2)+ x(3)x(4) + x(4)(2)) + l(x(5)(2) + x(5)x(6) + x(6)(2) + x(7)(2) + x(7)x(8) + x(8)(2)) for (k, l) = (1, 16), (1, 18), (2, 9). A modular form approach is used.