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  1. ; ; (, Proceedings of the National Academy of Sciences)
    Ribet, Kenneth (Ed.)
    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. 
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  2. ; (, Journal of Number Theory)
    It was conjectured by physicists that a particular shifted sum of even divisor sums vanishes, and a formal argument was later given for this vanishing. Shifted convolution sums of this form appear when computing the Fourier expansion of coefficients for the low energy scattering amplitudes in type IIB string theory and have applications to subconvexity bounds of L-functions. In this article, we generalize the argument from and rigorously evaluate shifted convolution of the divisor functions. In doing so, we derive exact identities for these sums and conjecture particular identities similar to but different from the one originally found. 
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  3. ; (, Communications in Number Theory and Physics)
    In this article, we find the full Fourier expansion for solutions of a certain differential equation. When such an f is fully automorphic these functions are referred to as generalized non-holomorphic Eisenstein series. We give a connection of the boundary condition on such Fourier series with convolution formulas on the divisor functions. Additionally, we discuss a possible relation with the differential Galois theory. 
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