Convolution identities for divisor sums and modular forms

We consider certain convolution sums that are the subject of a conjecture by Chester, Green, Pufu, Wang, and Wen in string theory. We prove a generalized form of their conjecture, explicitly evaluating absolutely convergent sums [Formula: see text]where [Formula: see text] is a Laurent polynomial with logarithms. Contrary to original expectations, such convolution sums, suitably extended to [Formula: see text], do not vanish, but instead, they carry number theoretic meaning in the form of Fourier coefficients of holomorphic cusp forms.