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On some positivity properties of the Riemann Ξ\Xi function

A mix of human-written and AI-generated textHuman understanding: all partsmath.CA — Classical Analysis and ODEsmath.NT — Number Theory

Contributed by Suvrit Sra ↗

Version 1 / Oct 27, 2026 / CC BY 4.0

Abstract

We prove strict log-concavity of the positive functions Θ\Theta and Λ\Lambda with one-sided and bilateral Laplace transforms 1/ξ(12+s)1/\xi(\tfrac12+\sqrt s) and 1/Ξ1/\Xi, respectively, where Ξ(z)=ξ(12+iz)\Xi(z)=\xi(\tfrac12+iz). For either function, log-concavity is the Pólya frequency property of order two (PF2_2). Without assuming RH, but using known zero verification through height 3⋅10123\cdot10^{12}, we further prove that Λ\Lambda is a PF function of order ten (PF10_{10}). We also prove local PF positivity for Θ\Theta: for each fixed order, the ordered minors of Θ(x−y)\Theta(x-y) are positive when all differences xi−yjx_i-y_j are sufficiently small and positive. Additionally, we prove the generalized Laguerre inequalities: if RH holds up to height T>1/2T>1/2, then Ln[Ξ](x)≥0L_n[\Xi](x)\ge0 for every n≥0n\ge0 and ∣x∣≤T−1/2|x|\le T-1/2; the case n=1n=1 is due to Csordas, Ruttan and Varga (1991).

Provenance statement

The results in this paper were developed through extensive use of an AI assistant (OpenAI's GPT Pro). None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs. The reciprocal-density revisions, verification programs, and attribution review also used OpenAI Codex. Anthropic's Claude (Opus 5.5 and Fable 5.1) was used extensively to assist with the writing. Assistant checks and program reruns are not external peer review or formal verification.

Tools used

OpenAI
GPT Pro
OpenAI
Codex
Anthropic
Claude Opus 5.5 and Fable 5.1

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Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 27, 2026