Convolution identities for divisor sums and modular forms - Fedosova, Ksenia; Klinger-Logan, Kim; Radchenko, Danylo (Proceedings of the National Academy of Sciences - PNAS, 2024)
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 ∑ n 1 ∈ Z ∖ { 0 , n } φ ( n 1 , n − n 1 ) σ 2 m 1 ( n 1 ) σ 2 m 2 ( n − n 1 ) , where φ ( n 1 , n 2 ) is a Laurent polynomial with logarithms. Contrary to original expectations, such convolution sums, suitably extended to n 1 ∈ { 0 , n } , do not vanish, but instead, they carry number theoretic meaning in the form of Fourier coefficients of holomorphic cusp forms.