Evaluation of the Convolution Sums ∑l+15m=nσ(l)σ(m) AND ∑3l+5m=nσ(l)σ(m) And an Application

Abstract

We evaluate the convolution sums ∑l,m∈ℕ,l+15m=nσ(l)σ(m) and ∑l,m∈ℕ,3l+5m=nσ(l)σ(m) for all n ∈ ℕ using the theory of quasimodular forms and use these convolution sums to determine the number of representations of a positive integer n by the form

We also determine the number of representations of positive integers by the quadratic form

by using the convolution sums obtained earlier by Alaca, Alaca and Williams [Evaluation of the convolution sums ∑l+6m=nσ(l)σ(m) and ∑2l+3m=nσ(l)σ(m), J. Number Theory124(2) (2007) 491–510; Evaluation of the convolution sums ∑l+24m=nσ(l)σ(m) and ∑3l+8m=nσ(l)σ(m), Math. J. Okayama Univ.49 (2007) 93–111].