Published October 17, 2026 | Version v2
Preprint Open

Ramanujan's prime-counting inequality holds for all x ≥ 38,358,837,683

Authors/Creators

  • 1. EDMO icon Purdue University

Description

Ramanujan’s prime counting inequality, unconditionally

Let N_R be the smallest positive integer such that Ramanujan’s inequality π(x)² < (ex/log x)·π(x/e) holds for every real x ≥ N_R. Assuming the Riemann hypothesis, Dudek and Platt showed that N_R = 38 358 837 683. This paper proves the same value unconditionally.

Earlier unconditional results left the range e^103 < x < e^3158.442 uncovered, and this paper closes it. The proof uses only the verification of the Riemann hypothesis up to height 3·10^12 (Platt–Trudgian), together with:

  • a smoothed explicit formula for ψ(x), of Büthe type;
  • the Bellotti–Trudgian–Yang zero-free region, evaluated exactly at each height below e^43;
  • a repulsion bound for pairs of zeros close to the line σ = 1;
  • a sharpened version of Bellotti’s log-free zero-density estimate. Its main new ingredient is an asymptotically sharp explicit bound for the Barban–Vehov mean square, proved in full in Appendix A.

Every numerical step is certified in ball (interval) arithmetic using Arb/python-flint. Floating point is used only to choose parameters.

Contents of this record

  • ramanujan-inequality.pdf: the paper (25 pages)
  • ramanujan-inequality-source.tar.gz: the LaTeX source
  • a code snapshot with all certification scripts and their logs; certify/arb/run_all.sh reproduces the verification

Changes in v2

  • The text is rewritten for readability.
  • Appendix A, the complete proof of the Barban–Vehov bound (Theorem G), is new.
  • Several remainder terms in that bound are corrected, and the whole computation has been recertified. The final margins are max q = 0.99137 on [57, 3159) and transfer ratio 0.99052.
  • The paper now states explicitly what it uses from Bellotti–Trudgian–Yang.
  • Notation is cleaned up and the bibliography is updated.
  • The main result is unchanged.

Links

AI disclosure: the research and writing were done with substantial assistance from Claude (Anthropic), as described in the paper’s acknowledgement.

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Additional details

Related works

Is identical to
Preprint: https://aixiv.online/abs/2610.01760 (URL)
Is new version of
Preprint: 10.5281/zenodo.23408098 (DOI)
Is supplemented by
Software: https://github.com/jaideepsaipadhi/ramanujans-inequality (URL)

Dates

Available
2026-10-17
Preprint

Software

Repository URL
https://github.com/jaideepsaipadhi/ramanujans-inequality
Programming language
Python
Development Status
Active

References

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