bibkey: ramare2008bruntitchmarsh authors: Oliver Ramaré and Jan Christoph Schlage-Puchta year: 2008 title: Improving on the Brun-Titchmarsh Theorem doi: null url: https://arxiv.org/abs/2607.04283v1 claim: A uniform interval upper bound for actual primes pays the local first-layer inventory capacity used with the retained integer-competitor concavity cost. strata_touched: [] license: citation-only triage: anchor

A uniform interval upper bound, directly reused

The primary is Improving on the Brun-Titchmarsh Theorem, arXiv:2607.04283v1, Theorem 1.1, PDF page 2 of 19. Its archive cover identifies the paper as published in Acta Arithmetica 131(4) (2008), 351–366. The arXiv version was uploaded in July 2026; the sieve theorem is a 2008 result, not a new 2026 prime-distribution improvement.

The downloaded PDF has SHA-256 bc3aab7b452a7598506e1c123724a6626cc9dc12c1b667c07a6e99faa211b5ef. These facts identify the inspected statement; the full sieve proof and its computations were not independently reproduced or Lean-verified.

The theorem states that there is $N_0$ such that

$$\pi (M+N)-\pi (M)\le \frac{2N}{\log N+3.53}(N\ge N_0,M\ge 1).$$

Its uniformity in the starting point is essential. The project uses outward integer rounding, so it requires no extension of the theorem to a new class of real input parameters. The start $N_0$ is existential in this statement; no numerical value is supplied here. There is no RH assumption, signed prime-error conclusion, interval asymptotic or almost-all exceptional-center selection in this input.

The existing price-gap application uses it for first-layer activation primes in $(B-1,B+z)$ when $z\ge \sqrt{B}$. After weighting each actual prime by its own $\log p$, the uniform capacity coefficient may be any fixed $c>4$, with the complete higher-layer contribution paid by an existing $O(\sqrt{B})$ frontier bound. The subsequent $8/5+\eta$ coefficient comes from that application’s retained work-plus-concavity account; it is not a theorem attributed to this paper.