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,l+15m=nσ(l)σ(m) and l,m,3l+5m=nσ(l)σ(m) for all 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 x12 + x1x2 + x22 + x3 2 + x3x4 + x42 + 5 (x52 + x5x6 + x6 2 + x72 + x7x2 + x 82). We also determine the number of representations of positive integers by the quadratic form x12 + x 22+x32+x42 + 6 (x52+x62+x7 2+x82), 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 Theory 124(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].