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, which were proved by Weil for curves and later, by Deligne for varieties over finite fields. 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, which uses the Frobenius operator on a finite field. The zeta function is then expressed as a determinant, allowing the properties of the function to relate to those of the operator. The search for a suitable cohomology theory and associated operator to prove the Riemann hypothesis is still on. In this paper, we study the properties of the derivative operator $D = \frac{d}{dz}$ on a particular weighted Bergman space of entire functions. The operator $D$ can be naturally viewed as the `i 查看全文>>