The Riemann Hypothesis: A Hilbert-Schmidt Operator

Presenting "The Analyst's Problem: A Hilbert-Schmidt Operator"...This paper develops a functional-analytic framework for encoding the smoothed Dirichlet energy associated with the Riemann zeta function into a single compact, self-adjoint operator on a separable Hilbert space. The construction relies on a rapidly decaying hyperbolic (sech-type) kernel that enforces positivity, symmetry, and Hilbert–Schmidt regularity, ensuring the resulting operator is well-defined and compact in finite-dimensional truncations. The central idea is to reinterpret a Toeplitz-type quadratic form arising from smoothed Dirichlet polynomials as the action of a structured infinite-dimensional operator acting on square-summable sequences. This operator admits consistent finite-dimensional truncations, which allow numerical stability checks and spectral diagnostics. A key contribution is the identification of a structured feature-map representation that renders the kernel positive semidefinite via a Bochner-type argument, together with 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 that is equivalent to the Riemann Hypothesis under standard assumptions on the explicit formula. The work is primarily structural: it establishes the analytic properties of the operator model, proves its compactness and self-adjointness in the appropriate sense, and reformulates a central number-theoretic conjecture as a spectral positivity problem.