A Dirichlet-Form Rewriting of the Weil Explicit Formula and an Unconditional Hilbert–Pólya Candidate

Abstract

For $\lambda > 1$, let $I_\lambda = [\lambda^{-1}, \lambda]$ and $\mathcal{H}_\lambda = L^2(I_\lambda, d^\times u)$. We show that the boundary-free semilocal Weil quadratic form on $\mathcal{H}_\lambda$, after an explicit scalar shift determined by the von Mangoldt function and by the digamma function at $1/4$, extends to a nonnegative closed symmetric Dirichlet form $(\mathcal{E}_\lambda, \mathcal{F}_\lambda)$. It decomposes canonically as the sum of a continuous pure-jump archimedean energy with explicit L\'evy measure $\nu(dr) = \pi^{-1}(1-e^{-2r})^{-1}e^{-r/2}\,dr$ and a finite arithmetic part consisting of dilation-induced discrete jumps and a prime-power killing term indexed by $n \le \lambda^2$. The form is irreducible, its self-adjoint generator $L_\lambda$ has compact resolvent, and the unique $L^2$-normalized ground state $\xi_\lambda$ is strictly positive almost everywhere and even under $u \mapsto u^{-1}$. The Fourier--Mellin transform of $\xi_\lambda$, and of every even-Galerkin approximant $\xi_{\lambda,N}$, is an entire function of exponential type whose zeros are all real, unconditionally on the Riemann Hypothesis. A zeta-regularized determinant identity identifies the zero set of $\widehat{\xi_{\lambda,N}}$ with the spectrum of an explicit finite self-adjoint operator, thereby furnishing an unconditional candidate in the spirit of Hilbert--Pólya.

Remark

The author will divide the unconditional complete proof of the Riemann Hypothesis into two papers. The one shown here is the first paper, which provides the analytical object for the base state approximation problem of the second paper. The second paper needs further improvement and will be published in subsequent work.