Evaluation of the convolution sums Sigma(l+27m=n) sigma(l)sigma(m) and Sigma(l+32m=n) sigma(l)sigma(m)

Abstract

We determine the convolution sums Sigma(l+27m= n) sigma(l)sigma(m) and Sigma(l+32m= n) sigma(l)sigma(m) 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 x(1)(2) + x(1)x(2) + x(2)(2) + x(3)(3) + x(3)x(4) + x(4)(2) + 9(x(5)(2) + x(5)x(6) + x(6)(2) + x(7)(2) + x(7)x(8) + x(8)(2)) and x(1)(2) + x(2)(2) + x(3)(2) + x(4)(2) + 8(x(5)(2) + x(6)(2) + x(7)(2) + x(8)(2)). A modular form approach is used.