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Award ID contains: 2001909

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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. ; (, Research in Number Theory)
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  4. ; (, 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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  5. ; ; (, Mathematische Zeitschrift)
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  6. (, Notices of the American Mathematical Society)
    Accepted Manuscript Full Text Available
  7. ; (, Journal of Number Theory)
    We correct an error found in Section 3.4.1 of Linear operators, the Hurwitz zeta function and Dirichlet L-functions published in JNT 217 (2020) 422–442. The error is related to the convergence of the inverse operator G−1 defined in Section 3.3 and affects the statement and proof of Proposition 17. We provide a revised statement and proof. 
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  8. ; ; ; ; ; (, Taylor & Francis)
    We detail our experience running an annual four-day workshop at the University of Minnesota, called the Mathematics Project at Minnesota (MPM). The workshop is for undergraduates who come from groups underrepresented in mathematics and aims to increase the participation and success of such groups in the mathematics major at the University. In this paper, we explain how MPM is organized, discuss its objectives, and highlight some of the sessions that we feel are emblematic of the program's success. The paper concludes with an analysis of achievements and obstacles in the programs' first three years. 
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  9. ; (, Journal of Number Theory)
    At the 1900 International Congress of Mathematicians, Hilbert claimed that the Riemann zeta function is not the solution of any algebraic ordinary differential equation its region of analyticity. In 2015, Van Gorder addresses the question of whether the Riemann zeta function satisfies a non-algebraic differential equation and constructs a differential equation of infinite order which zeta satisfies. However, as he notes in the paper, this representation is formal and Van Gorder does not attempt to claim a region or type of convergence. In this paper, we show that Van Gorder's operator applied to the zeta function does not converge pointwise at any point in the complex plane. We also investigate the accuracy of truncations of Van Gorder's operator applied to the zeta function and show that a similar operator applied to zeta and other L-functions does converge. 
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