Convolution identities for divisor sums and modular forms
Abstract
We prove exact identities for convolution sums of divisor functions of the form Σn1 ∈ Z \0,n\(n1,n-n1)σ2m1(n1)σ2m2(n-n1) where (n1,n2) is a Laurent polynomial with logarithms for which the sum is absolutely convergent. Such identities are motivated by computations in string theory and prove and generalize a conjecture of Chester, Green, Pufu, Wang, and Wen from CGPWW. Originally, it was suspected that such sums, suitably extended to n1∈\0,n\ should vanish, but in this paper we find that in general they give Fourier coefficients of holomorphic cusp forms.
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