bibkey: yamada2023arbitraryintervals
authors: Tomohiro Yamada
year: 2023
title: Explicit improvements of the Brun-Titchmarsh theorem for arbitrary intervals
doi: null
url: https://arxiv.org/pdf/2312.16090v1
claim: Theorem 2 gives an explicit uniform prime count for arbitrary real intervals; its modulus-one case supplies the finite thin-layer capacity in the existing CA-benefit application.
strata_touched: []
license: citation-only
triage: anchor

A uniform finite interval input for the CA benefit

The inspected source is the preprint arXiv:2312.16090v1, Theorem 2, equation (13), printed p.3. For relatively prime integers $k\ge 1$ and $b$, and every real $x>0$, $y>k$, it states

$$\pi (x+y;k,b)-\pi (x;k,b)<\frac{2y}{\phi (k)(\log (y/k)+0.8601)}.$$

In particular $k=1$ gives a bound at every interval centre, without an existential large-length threshold. The application uses only the weaker $2y/\log y$ consequence. The preprint’s proof, numerical calculations, and full general-modulus claim have not been independently verified or formalized in Lean. No strongest-available or publication-priority claim is made here.

The existing benefit application uses it at $x=(1-2h)a>0$, $y=4ha\ge 4\sqrt{a}$, for $a\ge 16$ and $a^{-1/2}\le h\le 1/4$. Every counted prime keeps its own logarithmic weight. Smaller $h$ is bounded by monotonicity; the complete higher-layer mass is paid separately by the cited Dusart estimate. This supplies an explicit version of an already used square-root-buffer mechanism, not a new short-interval theorem or a signed Robin estimate.