The Riemann Hypothesis: A Hilbert Operator

Presenting "The Analyst's Problem: Hilbert Operator"...This paper develops a functional-analytic framework for encoding smoothed Dirichlet energy associated with the Riemann zeta function into a single compact, self-adjoint operator on a separable Hilbert space. The construction is based on a rapidly decaying hyperbolic kernel that enforces positivity, symmetry, and Hilbert–Schmidt regularity, ensuring the resulting operator is well-defined, compact, and spectrally stable. The central idea is to reinterpret a Toeplitz-type quadratic form arising from smoothed Dirichlet polynomials as the action of a single infinite-dimensional operator acting on square-summable sequences. This operator is shown to admit consistent finite-dimensional truncations, allowing numerical stability checks and spectral diagnostics that converge rapidly. A key contribution is the identification of a structured feature-map representation that renders the kernel positive semidefinite via a Bochner-type argument, alongside a weighted finite-dimensional embedding that enforces exponential decay. The framework connects operator theory with analytic number theory by linking the positivity of the quadratic form in the infinite limit to a classical criterion equivalent to the Riemann Hypothesis under standard explicit-formula assumptions. The work is primarily structural: it establishes a coherent operator model, proves its analytic properties, and reformulates a central number-theoretic conjecture as a spectral positivity problem.