Abstract
This essay examines the structural similarity between the asymptotic formula for the proportion of permutations fixing a k-set, recently established by Green and Sawhney, and the heat-flow deformation theory of the Riemann xi-function associated with the de Bruijn–Newman constant.
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Source Paper and Its Core Structure
Green and Sawhney study the limiting probability p(k) that a random permutation on n letters has an invariant set of size k. Their main result expresses p(k) asymptotically as f({log₂ k}) k⁻ᵟ (log k)⁻³ᐟ², where δ ≈ 0.086 is the Erdős–Ford–Tenenbaum constant and f is a smooth periodic function on the circle. Remarkably, f is almost constant, with max f / min f < 1 + 2×10⁻⁷. The function f arises as a triple convolution c₀ g ∗ μ ∗ μ′, where g is an explicit infinite series involving powers of 2, and μ, μ′ are measures defined via expectations of random walk statistics (ϱ and τ) associated with Poisson processes.
The de Bruijn–Newman Connection
The de Bruijn–Newman constant Λ governs the evolution of the Riemann xi-function under a heat flow: for t ≥ Λ, the deformed function ξₜ has only real zeros. This essay speculates that the convolution structure (g ∗ μ ∗ μ′) in the permutation problem serves as a discrete analogue of this heat-flow smoothing. The function g plays the role of the heat kernel, its Fourier coefficients decaying rapidly via values of the Gamma function at imaginary arguments, analogous to the spectral multipliers in the continuous heat flow. The near-constancy of f mirrors the "stability" or reality of zeros expected in the de Bruijn–Newman framework.
Assessment and Limitations
The proposed analogy is rated as a suggestive metaphor. While both contexts exhibit rapid Fourier decay mediated by the Gamma function, the source domain lacks the functional equation and Euler product essential to the Riemann Hypothesis. Moreover, the source convolution is a static averaging procedure, whereas the de Bruijn–Newman flow is a continuous time evolution. The essay concludes with proposed numerical experiments to test the spectral properties of the measures μ and μ′, and an analysis of why the analogy fundamentally breaks down as a tool for attacking RH.
This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.