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Ramanujan's prime counting inequality holds for all x ≥ 38,358,837,683: ranges from the zeros, a spectral barrier, and trade-offs

Primarily AI-generated textHuman understanding: all partsmath.NT — Number Theory

Contributed by Sayed Barez ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

Ramanujan proved that π(x)2<exlog⁡xπ(x/e)\pi(x)^2 < \frac{ex}{\log x}\pi(x/e) for all sufficiently large xx. Let NRN_R be the least integer such that this holds for every real x≥NRx \ge N_R. Dudek and Platt showed that the Riemann hypothesis implies NR=38,358,837,683N_R = 38{,}358{,}837{,}683. We prove that NR=38,358,837,683N_R = 38{,}358{,}837{,}683. To our knowledge, this is the first determination of NRN_R without assuming the Riemann hypothesis. The proof uses the zero-free half-plane Res>7/8\mathrm{Re}s > 7/8 of a recent OpenAI preprint. Its main statement for ζ\zeta is formalized in Lean 4 and has been rebuilt independently; its written proof has not been peer reviewed. The deduction is our Theorem B: if ζ(s)≠0\zeta(s) \ne 0 for Res>0.945092\mathrm{Re}s > 0.945092, then NR=38,358,837,683N_R = 38{,}358{,}837{,}683. The proof of Theorem B uses only refereed inputs, together with Bhattacharjee's correction of a theorem of Büthe. Relying only on the peer-reviewed literature, and on preprints where stated, we also prove the following by rigorous computer-assisted interval arithmetic. (A) The inequality holds for 38,358,837,683≤x≤e753.9838{,}358{,}837{,}683 \le x \le e^{753.98}. The previous range ended at e103e^{103}. (C) It holds for x≥e3035.01x \ge e^{3035.01}. This replaces Axler's threshold e3158.442e^{3158.442}, whose proof applies a lemma of Dudek and Platt in a case where that lemma's proof is defective. (B′, D) Either of the following also suffices: a zero-free region β≤1−1/(1.7373log⁡∣γ∣)\beta \le 1 - 1/(1.7373\log|\gamma|) above height 3⋅10123\cdot10^{12}, or RH verified to height 4.1602⋅10144.1602\cdot10^{14}. We tabulate the trade-off between the two. (E) Arguments that use only non-negativity, analytic continuation and the poles up to height TT cannot go beyond e754.40e^{754.40} when T=3⋅1012T = 3\cdot10^{12}. So (A) is optimal among such arguments to within 0.41 in log⁡x\log x. Building on Bhattacharjee's discovery that the constant E2E_2 in Büthe's Theorem 1 lacks a factor log⁡(3c)\log(3c), we show that the affected rows of the Broadbent–Kadiri–Lumley–Ng–Wilk table inherit the error, by 1.3–2.2% after re-optimization. We also trace the consequences in the literature.

Provenance statement

This paper was produced by the author directing AI agents (Claude, Anthropic, via Claude Code) that carried out the literature search, the mathematical derivations, the computer-assisted verification and the drafting of the manuscript. All numerical claims are produced by the Python scripts in anc/. Every inequality used in a proof is checked in outward-rounded interval arithmetic (mpmath), on grids with adaptive bisection, and every number in the paper is taken from a saved output file. The author's AI agents independently re-checked key computations: the reproduction of published tables, a quadrature cross-check of the Büthe constant, the reproduction of Johnston–Yang's and Lichtman–Pomerance's constants, and a regeneration of all final outputs from the final scripts. A separate AI agent also audited an earlier draft, and its corrections were incorporated. No human expert has refereed the arguments. Theorem B and Corollary F rely on OpenAI's zero-free half-plane Re s > 7/8 (preprint, 30 Sept 2026), whose statement for ζ has a Lean 4 formalization that was independently rebuilt by a third party; we did not rebuild it ourselves. The full repository, with all scripts and outputs, is at https://github.com/blueberryvertigo/ramanujan-prime-inequality-and-missing-factor.

Tools used

Anthropic
Claude OpusVersion 5.5
n/a
Python 3 + mpmath
n/a
NumPy / SciPy

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Version history

  1. v1Submitted by Sayed BarezInitial depositCurrentOct 19, 2026