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The Derivative Operator on Weighted Bergman Spaces and Quantized Number Theory


Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil conjectures. Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, relying on the Frobenius operator on the finite field. The zeta function is then expressed as a determinant allowing the properties of the function to relate to the properties of the operator. The search for a suitable cohomology theory and associated operator to prove the Riemann hypothesis has continued to this day. In this dissertation we study the properties of the derivative operator on a particular family of weighted Bergman spaces. This operator is meant to be the replacement for the Frobenius in the general case and is first used to give a method of quantizing elliptic curves and modular forms; then to construct an operator associated to any given meromorphic function. With this construction, we show that for a wide class of functions, the function can be recovered using a regularized Berezinian determinant involving the operator constructed from the meromorphic function. This is shown in some special cases: rational functions, zeta functions of algebraic curves over finite fields, geometric zeta functions of lattice self-similar strings, the gamma function, the Riemann zeta function and culminating in a quantized version of the Hadamard factorization theorem that applies to any entire function of finite order. This shows that all of the information about the given meromorphic function is encoded into the special operator we constructed.

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