LAUSR

The generating functions of divisor functions are quasimodular forms of weight 2 and the product of them is a quasimodular form of higher weight. In this work, we evaluate the convolution sums ∑a1m1+a2m2+a3m3+a4m4=nσ(m1)σ(m2)σ(m3)σ(m4) for the positive integers a1,a2,a3,a4, and n with lcm(a1,a2,a3,a4) ≤ 4. We reprove the known formulas for the number of representations of a positive integer n by each of the quadratic forms ∑j=016x j2 and ∑ j=18(x 2j−12 + x 2j−1x2j + x2j2) as an application of the new identities proved in this paper.