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A sharper Type I/II estimate for primes in arithmetic progressions to smooth moduli, and H_m << e^{3.8043 m}

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

In Stadlmann's Type I/II equidistribution estimate for x^delta-smooth moduli, one of the three conditions, 64 varpi + 20 delta + 2 sigma < 1, can be replaced by 48 varpi + 18 delta + 2 sigma < 1. This weaker condition follows from the second one, 48 varpi + 16 delta + 4 sigma < 1, whenever delta <= sigma. The proof follows Stadlmann's argument and makes two changes: the length of a block in one Taylor expansion depends on the pair of shifts, and the bounds are summed over the pairs instead of being multiplied by their number. Combined with Stadlmann's general Harman-sieve construction, this gives a minorant for the primes with exponent of distribution 1/2 + 2 varpi to smooth moduli for every varpi < 7/544, instead of varpi = 0.01265. As a consequence, liminf (p_{n+m} - p_n) << e^{cm} for every c > 3.80428, and for every c > 3.80423 if a numerically computed value of the sieve deficit is used. This improves Stadlmann's exponent 3.8075.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews against the cited articles and by computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

  1. R. C. Baker, A. J. Irving, Bounded intervals containing many primes, Math.\ Z. 286 (2017), 821--841.
  2. OpenAI, Improved short gaps between primes, preprint, 30 August 2026, https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf.link
  3. D. H. J. Polymath, New equidistribution estimates of Zhang type, Algebra Number Theory 8 (2014), 2067--2199.
  4. J. Stadlmann, On primes in arithmetic progressions and bounded gaps between many primes, Adv.\ Math. 468 (2025), 110190; arXiv:2309.00425v3.arXiv
  5. J. Stadlmann, Bounded gaps between primes, arXiv:2608.31126v1 (2026).arXiv

Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 03, 2026