arXiv2026
This note takes the probabilistic half of the Riemann Xi story on its own terms. Every object in the subject is a first passage time: Riemann's kernel is the law of the logarithm of a sum of two hitting times of a three dimensional Bessel process, Polya's approximation is the first passage of a Brownian motion with drift, and the reciprocal Xi function, under the Riemann hypothesis, is the Laplace transform of an infinite convolution of exponentials whose rates are the squared zeros. That reciprocal is written as $F_α(s)=ξ(α)/ξ(α+\sqrt s)$, and complete monotonicity, unconditional for $α\ge1$, is conjectured to persist to the critical basepoint $α=1/2$. Kent's eigenvalue expansion says which laws can arise this way, namely those whose Thorin measure is a Dirichlet spectrum with unit atoms, and Krein's inverse spectral theory turns the hypothesis into the existence of a string. The passage from Riemann to Polya is a flow, not a jump: the Cauchy semigroup on Thorin measures, each step an Esscher tilt followed by a Brownian subordination along a curvature family of hyperbolic Bessel processes, with the arithmetic surviving as Fourier modes damped like $e^{-2πk\varepsilon}$. The arithmetic lives in the atoms and nowhere else. Approximations rank by what they keep: Polya keeps neither atoms nor tempering and is off by a factor of three, a fitted Bessel dimension reaches one per cent, and a few atoms with an erfc tempering stay better than one part in a thousand across four decades. And the flow runs backwards: the Thorin measure of the reciprocal is evaluated from a prime sieve with no reference to any zero, and nonnegative deconvolution of it returns the first ten zero ordinates with unit masses, nine of them to four decimals and one to three. All identities are verified with mpmath, and the scripts are included.