On the number of representations of an integer by certain quadratic forms in sixteen variables

Abstract

We evaluate the convolution sums ∑l,m∈ℕ,l+2m=n σ3(l)σ3(m), ∑l,m∈ℕ,l+3m=n σ3(l) × σ3(m), ∑l,m∈ℕ,2l+3m=n σ3(l)σ3(m) and ∑l,m∈ℕ,l+6m=n σ3(l)σ3(m) for all n ∈ ℕ using the theory of modular forms and use these convolution sums to determine the number of representations of a positive integer n by the quadratic forms Q8 ⊕ Q8 and Q8 ⊕ 2Q8, where the quadratic form Q8 is given by x21 + x1x2 + x22 + x23 + x3x4 + x24 + x25 + x5x6 + x26 + x27 + x7x8 + x28.