bibkey: maynard2013bruntitchmarsh authors: James Maynard year: 2013 title: On the Brun-Titchmarsh theorem doi: 10.4064/aa157-3-3 url: https://arxiv.org/pdf/1201.1777v2 claim: Published Theorem 3.2 supplies an effective uniform lower prime count for q at least q0, reduced classes and x at least q to the eighth power; its negative-class sum is reused in the linked, branch-restricted actual-CA application. strata_touched: [] license: citation-only triage: anchor

Pointwise progression counts for the actual CA capacity

The inspected primary sources are the published article, Acta Arithmetica 157 (2013), no.3, 249–296, DOI:10.4064/aa157-3-3, and arXiv:1201.1777v2, dated 21 May 2012. The lower count is published Theorem 3.2, printed p.251, corresponding to manuscript Theorem 2, printed p.3. Published equations (3.8) and (3.11), printed pp.253–254, give the exceptional-zero and ordinary cases and the effective constants. The theorem is reused, without reproducing its analytic proof. No independent full proof audit, Lean verification, or strongest-available-result claim is made.

The published lower count

There are absolute effective constants $q_0\ge 3$ and $c_0>0$ such that every integer $q\ge q_0$, every reduced class $a\mathrm{mod}q$, and every real $x\ge q^8$ satisfy

$$\pi (x;q,a)\ge c_0\frac{\log q}{\sqrt{q}}\frac{x}{\phi (q)\log x}.\text{\hspace{1em}(M1)}$$

This is published Theorem 3.2 (manuscript Theorem 2), with its implied constant written as $c_0$. No GRH, exclusion of exceptional real zeros, prime-modulus hypothesis or bounded cubic-part condition is imposed. Published Theorem 3.3 has a different, ineffective $q^{-\epsilon }$ factor; it is not substituted for (M1). Published Theorem 3.1 is an upper bound and is not used as the required lower supply.

Let $\chi$ be any real nonprincipal Dirichlet character modulo $q$, primitive or imprimitive, and define

$${\pi }_{-}(x;\chi )=\#\{p\le x:\chi (p)=-1\}.$$

On the unit group, $\chi$ is a surjection to $\{1,-1\}$, so exactly $\phi (q)/2$ reduced classes have value $-1$. Those classes are disjoint. Summing the same uniform lower bound (M1), rather than giving each class an independent copy of the prime population, yields

$$\boxed{{\pi }_{-}(x;\chi )\ge \frac{c_0}{2}\frac{x\log q}{\sqrt{q}\log x}}(q\ge q_0,x\ge q^8).\text{\hspace{1em}(M2)}$$

Primes dividing $q$ remain zeros and occur in none of these classes. This is an application of the published theorem, not a new analytic estimate or an originality claim.

The actual-source application uses (M2) at $x=P^{+}(n)$, comparing it with the number of primes whose CA exponent could be even at that same integer. It needs no promotion of an unweighted prime count to reciprocal-prime mass. Conductors with $q^8>P^{+}(n)$ are outside this supplier’s range; it gives no signed Robin estimate there.