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Abstract
In this thesis we present a new method to construct modular forms using rational functions. It relies on contour integration and Weil's converse theorem. We give several applications, reaching from a relation between cotangent sums and $L$-functions, formulas for Eichler integrals and period polynomials and series representations for $L$-functions corresponding to products of Eisenstein series. \\ With similar ideas, based on contour integration, we move on to equations which were originally studied by Ramanujan and generalize his formulas to those containing $L$-functions at rational arguments. We work out a very general framework for finding new equations of the Ramanujan type that can be applied to a wide range of $L$-functions.
Document type: | Dissertation |
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Supervisor: | Kohnen, Prof. Dr. Winfried |
Place of Publication: | Heidelberg |
Date of thesis defense: | 10 September 2020 |
Date Deposited: | 17 Sep 2020 09:07 |
Date: | 2020 |
Faculties / Institutes: | The Faculty of Mathematics and Computer Science > Institut für Mathematik |
DDC-classification: | 510 Mathematics |
Controlled Keywords: | Modular forms, L-functions, Period polynomials |