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).