Operator-valued Euler products from abstract prime systems and rigidity

Walid OUKIL

doi:10.33774/coe-2026-tcgbf

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Operator-valued Euler products from abstract prime systems and rigidity

30 June 2026, Version 1

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Working Paper

Walid OUKIL

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This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

We introduce an abstract notion of prime system (a purely structural concept that captures factorisation without reference to ordinary integers) and construct an associated operator‑valued Euler product. The scalar differential equation of the rigidity programme is lifted to a Banach‑space setting, and the role of the non‑homogeneous term $\eta$ is played by a piecewise‑linear profile derived from the counting function of the abstract integers. For a normal operator $W$ with positive real part, the functional $\mu_\eta(W)$ is expressed through an operator‑valued zeta function, which admits an Euler product indexed by the abstract primes. Inverting this relation yields a direct link between the Euler product and $\mu_\eta(W)$, providing a tool for analysing the rigidity criterion $\mathcal{L}_\eta(W,\theta)\neq I$. The framework is general enough to encompass classical Beurling primes, graph‑theoretic prime nodes, or any factorisation semi‑group.

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About the Prime Rigidity Theory (PR): I am currently developing a long‑term research programme called the Prime Rigidity Theory (PR). This work builds on my arlier results concerning bounded solutions of complex differential equations and the rigidity inequalities that force structural asymmetry. The goal now is to generalise these ideas step by step to an operator‑theoretic framework, where the complex parameter is replaced by a bounded linear operator, and the classical Euler product is replaced by an operator‑valued product over an abstract prime system. Because the project is mainly a formal construction at this stage, I am using freely available AI platforms (with limited usage) as a day‑to‑day assistant. These tools help me draft and organise the working papers, but every definition, theorem, and proof is guided by the scalar theory alread established. The current writings are deliberately kept as “working papers”: they aim to build a clean formal skeleton of the theory, rather than to provide finished computational results. All the details are recoverable by specialising the operator statements to the classical scalar case. The vision is that the abstract prime systems, the operator‑valued Euler product, and the rigidity functionals together form a unified language that connects differential equations, functional calculus, and factorisation structures in a completely new way.