Evaluation of the convolution sum ∑al+bm=nσ(l)σ3(m) for (a,b)=(1,7),(7,1),(1,8),(8,1),(1,9),(9,1) and representations by certain quadratic forms in twelve variables

Abstract

For a positive integer n we evaluate the convolution sum Sigma(al+ bm=n) sigma(l) sigma(3)(m) for (a, b) = (1, 7), (7, 1), (1, 8), (8, 1), (1, 9), (9, 1) . We then use these evaluations together with knownevaluations of other convolution sums to determine the numbers of representations of n by the forms x(1)(2) + x(2)(2) + x(3)(2) + x(4)(2) + 2(x(5)(2) + x(6)(2) + x (2)(7) + x(8)(2) + x (2)(9) + x(10)(2) + x(11)(2) + x(12)(2)), x(1)(2) + x(2) (2) + x(3)(2) + x(2)(4) + x(5)(2) + x(6)(2) + x(7)(2) + x(8)(2) + 2(x(9)(2) + x(10)(2) + x(11)(2) + x(12)(2)), x(1)(2) + x(1)x(2) + x(2) (2) + x(3)(2) + x(3)x(4) + x(4)(2) + 3(x(5)(2) + x(5)x(6) + x(6)(2) + x(7)(2) + x(7)x(8) + x(8)(2) + x(9)(2) + x(9)x(10) + x(10)(2) + x(11)(2) + x(11)x(12) + x(12)(2)), x(1)(2) + x(1)x(2) + x(2)(2) + x(3)(2) + x(3)x(4) + x(4)(2) + x(5)(2) + x(5)x(6) + x(6)(2) + x(7)(2) + x(7)x(8) + x(8)(2) + 3(x(9)(2) + x(9)x(10) + x(10)(2) + x(11)(2) + x(11)x(12) + x(12)(2)). We use a modular form approach.