Title: 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].