Abstract intersection theory and operators in Hilbert space

TL;DR

This paper introduces an abstract intersection theory for certain Hilbert space operators and proves its equivalence to the Riemann Hypothesis, linking spectral properties of operators to one of mathematics' most famous conjectures.

Contribution

It establishes a novel axiomatic framework connecting operator theory in Hilbert spaces with the Riemann Hypothesis, providing a new perspective on this longstanding problem.

Findings

Abstract intersection theory is equivalent to the Riemann Hypothesis for specific operators.

The theory offers a new approach to analyze the zeros of the Riemann zeta-function.

If zeros originate from such operators, the Riemann Hypothesis holds true.

Abstract

For an operator of a certain class in Hilbert space, we introduce axioms of an abstract intersection theory, which we prove to be equivalent to the Riemann Hypothesis concerning the spectrum of that operator. In particular if the nontrivial zeros of the Riemann zeta-function arise from an operator of this class, the original Riemann Hypothesis is equivalent to the existence of an abstract intersection theory.