{"id":"hexagon/2610.00584","pdf":"https://hexagonmath.org/pdf/2610.00584v1","url":"https://hexagonmath.org/abs/2610.00584v1","tags":["math.NT","msc:11M26","msc:26A51","msc:15B48","msc:44A10","Polya frequency","Riemann Xi","complete monotonicity","generalized Laguerre","log-concavity"],"score":0,"title":"On some positivity properties of the Riemann $\\Xi$ function","domain":"math.CA","origin":"ai-generated","source":"hexagon","authors":["Suvrit Sra"],"reviews":0,"subject":"math.CA","summary":"We prove strict log-concavity of the positive functions $\\Theta$ and $\\Lambda$ with one-sided and bilateral Laplace transforms $1/\\xi(\\tfrac12+\\sqrt s)$ and $1/\\Xi$, respectively, where $\\Xi(z)=\\xi(\\tfrac12+iz)$. For either function, log-concavity is the Pólya frequency property of order two (PF$_2$). Without assuming RH, but using known zero verification through height $3\\cdot10^{12}$, we further prove that $\\Lambda$ is a PF function of order ten (PF$_{10}$). We also prove local PF positivity for $\\Theta$: for each fixed order, the ordered minors of $\\Theta(x-y)$ are positive when all…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["GPT Pro","Codex","Claude Opus 5.5 and Fable 5.1"],"discussed":0,"published":"2026-10-27","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}