bibkey: hayneswhite2014intervals authors: Alan Haynes and Christopher J. White year: 2014 title: Group automorphisms with prescribed growth of periodic points, and small primes in arithmetic progressions in intervals doi: 10.1016/j.aim.2013.11.014 url: https://arxiv.org/abs/1309.2562v1 claim: Theorem 1.5 gives a prime in every reduced progression inside a fixed-ratio interval beyond modulus exponent 13.4; its direct quadratic-character application gives terminal negative primes for every fixed conductor exponent below 5/67, with an ineffective onset. strata_touched: [] license: citation-only triage: anchor

A published progression interval with a numerical exponent

The article appeared in Advances in Mathematics 252 (2014), 572–585, DOI:10.1016/j.aim.2013.11.014. The inspected primary text is arXiv:1309.2562v1, Theorem 1.5 on printed p.4 and its proof in §3, especially (3.3) on p.9. This note reuses the published interval result; its analytic proof has not been reproduced or Lean-verified here. No optimality or complete literature-search claim is made.

For every fixed $\kappa >13.4$ and $\epsilon >0$, every sufficiently large modulus $m$, every $a$ with $(a,m)=1$, and every $x>m^{\kappa }$, Theorem 1.5 supplies a prime

$$p\equiv a(\mathrm{mod}m),x\le p<(1+\epsilon )x.\text{\hspace{1em}(H1)}$$

This is an interval theorem, rather than only a least-prime theorem. The exceptional-zero case of its proof uses Siegel’s theorem in (3.3). No effective onset is claimed for (H1) or the application below.

For a real nonprincipal primitive quadratic character $\chi$ of conductor $q$, choose a unit class $a$ with $\chi (a)=-1$. Fix $0<\eta <5/67$ and choose $13.4<\kappa <1/\eta$. Apply (H1) with

$$m=q,x=2X/3,\epsilon =1/2.$$

If $q\le X^{\eta }$, then $q^{\kappa }\le X^{\eta \kappa }<2X/3$ eventually, uniformly over this conductor range. The supplied prime lies in $[2X/3,X)\subset (X/2,X]$ and has $\chi (p)=-1$. The finitely many smaller moduli excluded by the source’s threshold are handled by fixed-modulus prime distribution and enlarging the common onset. Thus, for every fixed $0<\eta <5/67$, all sufficiently large $X$ and all such characters with $q\le X^{\eta }$ have a negative-character prime in $(X/2,X]$. In particular $\eta =1/14$ is admissible. The prime is coprime to $q$; ramified zeros are preserved.

The actual CA conductor application uses only this interval supply. The Thorner–Zaman application separately gives an effective positive conductor exponent and a count, without numerical values for its constants. Neither interface alone provides the large-conductor weighted budget needed by Robin.