Evaluation of four convolution sums and representation of integers by certain quadratic forms in twelve variables

Abstract

In this paper the convolution sums ∑6i+j=n σ(l)σ3(m), ∑2i+3j=n σ(l)σ3(m), ∑i+6j=n σ(l)σ3(m) and ∑3i+2j=n σ(l)σ3(m) are evaluated for all n∈N, and then their evaluations are used to determine the representation number formulae N(1,1,1,1,1,2;n),N(1,1,1,1,2,2;n) and N(1,1,1,2,2,2;n) where N(a1,...,a6;n) denote the representation numbers of n by the form a1(x21 + x1x2 + x22) + a2(x23 + x3x4 + x24 ) + a3(x25 + x5x6 + x26) + a4(x27 + x7x8 +x28 ) +a5(x29 +x9x10 +x210) +a6(x211 +x11x12 +x212).