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Some positivity properties of Riemann Ξ\Xi under smoothing

A mix of human-written and AI-generated textHuman understanding: some partsmath.CA — Classical Analysis and ODEsmath.NT — Number Theorymath.PR — Probability

Contributed by Suvrit Sra ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We show that smoothing the zeros of the Riemann Ξ\Xi-function makes it easy to establish positivity properties that are otherwise hard (as they are equivalent to the Riemann hypothesis). Specifically, we smooth each zero with a small density centered at it, and then test the impact of this smoothing on complete monotonicity (CM) of the heat sum H(t)=∑ρe−tzρ2H(t)=\sum_\rho e^{-tz_\rho^2} over the zeros 12+izρ\tfrac12+iz_\rho. A sufficiently wide density quickly implies CM of the smoothed heat sum. We bound the width (i.e., half-variance) the density needs using increasingly precise facts about the zeros. If we only use that they lie in the critical strip, half-variance 1/81/8 suffices; the primes lower this width to π2/288\pi^2/288; and zeros verified through height TT further lower it to π2/(8log⁡2T)\pi^2/(8\log^2T) asymptotically. Ultimately, however, smoothing seems to be of limited utility: despite the CM result, the smoothed reciprocal laws of Ξ\Xi are not even log-concave, unlike the unsmoothed ones.

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. Anthropic's Claude (Fable 5.1 and 5.2) helped with the writing and with verifying some of the mathematics. The main text has not yet been vetted line by line by the author.

Tools used

OpenAI
GPT Pro
Anthropic
Claude Fable 5.1 and 5.2

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Version history

  1. v1 · 35 pagesSubmitted by Suvrit SraInitial depositCurrentNov 11, 2026