An Operator-Theoretic Framework for the Riemann Hypothesis: Finite Certificates, Schur-Complement Structure, and Global Positivity Questions
Inv ehsan akbari fartkhouni
Abstract
This paper organizes an operator-theoretic research program aimed at the Riemann hypothesis. Its central theme is the translation of positivity questions associated with the completed zeta function into quadratic-form and operator inequalities, with finite-dimensional certificates serving as computational evidence and as a guide to the structure of the infinite-dimensional problem. The paper records a Schur-complement reduction, isolates a reference-energy obstruction by an exact rational counterexample, summarizes a reported local interval-arithmetic certificate at the parameter value Y = 7, and collects finite arithmetic and finite-element computations that have been used to probe the proposed framework.
The most important logical distinction is between positivity of a reduced quadratic form and a stronger comparison involving a separately defined reference energy. An explicit rational example shows that the former does not, by itself, imply the latter. This identifies a precise condition that any successful global argument must establish. The numerical material is stated with its scope and limitations: finite certificates concern specified finite matrices and parameter ranges; finite-element eigenvalue observations do not establish positivity in an infinite-dimensional limit. The purpose is to make the mathematical structure, reusable calculations, and remaining proof obligations accessible for independent scrutiny and further work.