Towards a fractal cohomology: Spectra of Polya--Hilbert operators, regularized determinants and Riemann zeros

Abstract

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 `infinitesimal shift of the complex plane'. Furthermore, this operator is meant to be the replacement for the Frobenius operator in the general case and is used to construct an operator associated to any suitable meromorphic function. We then show that the meromorphic function can be recovered by using a regularized determinant involving the above operator. This is illustrated in some important special cases: rational functions, zeta functions of curves over finite fields, the Riemann zeta function, and culminating in a quantized version of the Hadamard factorization theorem that applies to any entire function of finite order. Our construction is motivated in part by [23] on the infinitesimal shift of the real line, as well as by earlier work of Deninger [10] on cohomology in number theory and a conjectural `fractal cohomology theory' envisioned in [25] and [28].