Evaluation of Some Convolution Sums by Quasimodular Forms

Abstract

We evaluate the convolution sums ∑_{l+135m=n}σ(l)σ(m),  ∑_{3l+45m=n}σ(l)σ(m),  ∑_{5l+27m=n}σ(l)σ(m),∑_{9l+15m=n}σ(l)σ(m), ∑_{l+45m=n}σ(l)σ(m),∑_{5l+9m=n}σ(l)σ(m), and ∑_{3l+15m=n}σ(l)σ(m) for all nεN using the theory of quasimodular forms and use some of these convolution sums to determine the number of representations of a positive integer n by some direct sum of the forms x₲+xâ‚xâ‚‚+34x₂²,5x₲+5xâ‚xâ‚‚+8x₂²,4x₲±3xâ‚xâ‚‚+9x₂², 2x₲±xâ‚xâ‚‚+17x₂² of discriminant -135