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On Shifted Convolution Sums Involving the Fourier Coefficients of Theta Functions Attached to Quadratic Forms
Ravindran, Hari Alangat

2014, Doctor of Philosophy, Ohio State University, Mathematics.
The study of shifted convolution sums has acquired a prominent place in current number theory research owing to its potential applications to the sub-convexity problem, while quadratic forms have fascinated mathematicians since antiquity. This thesis, which deals with both these topics, studies shifted convolution sums involving the Fourier coefficients of Theta Series associated to a positive definite integral quadratic form and a cuspidal Hecke eigenform of integral weight. Our aim is to generalize the work of W. Luo, J. Hafner, and H. Iwaniec et al. in this new setting. Three independent approaches are used in this endeavour -- the spectral theory of the hyperbolic Laplacian, the δ-symbol method (a variant of the Hardy-Littlewood-Ramanujan circle method), and the theory of Poincaré series via a Poisson-Voronoï summation formula. We establish asymptotic formulae in all three aspects with the spectral theory approach providing the optimal estimate for the error term when one of the forms involved is cuspidal, while the δ-symbol method gives a sharp error term when only Theta Series are involved in the shifted convolution sum.
Wenzhi Luo, Dr. (Advisor)
Roman Holowinsky, Dr. (Committee Member)
Jeffery McNeal, Dr. (Committee Member)
Virginia Sanders, Dr. (Committee Member)
97 p.

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Ravindran, H. (2014). On Shifted Convolution Sums Involving the Fourier Coefficients of Theta Functions Attached to Quadratic Forms. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Ravindran, Hari. "On Shifted Convolution Sums Involving the Fourier Coefficients of Theta Functions Attached to Quadratic Forms." Electronic Thesis or Dissertation. Ohio State University, 2014. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

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Ravindran, Hari "On Shifted Convolution Sums Involving the Fourier Coefficients of Theta Functions Attached to Quadratic Forms." Electronic Thesis or Dissertation. Ohio State University, 2014. https://etd.ohiolink.edu/

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