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.

Fourier uniqueness in even dimensions.

In recent work, methods from the theory of modular forms were used to obtain Fourier uniqueness results in several key dimensions ([Formula: see text]), in which a function could be uniquely reconstructed from the values of it and its Fourier transform on a discrete set, with the striking application of resolving the sphere packing problem in dimensions [Formula: see text] and [Formula: see text] In this short note, we present an alternative approach to such results, viable in even dimensions, based instead on the uniqueness theory for the Klein-Gordon equation. Since the existing method for the Klein-Gordon uniqueness theory is based on the study of iterations of Gauss-type maps, this suggests a connection between the latter and methods involving modular forms. The derivation of Fourier uniqueness from the Klein-Gordon theory supplies conditions on the given test function for Fourier interpolation, which are hoped to be optimal or close to optimal.