bibkey: thornerzaman2019chebotarev authors: Jesse Thorner and Asif Zaman year: 2019 title: A unified and improved Chebotarev density theorem doi: 10.2140/ant.2019.13.1039 url: https://msp.org/ant/2019/13-5/ant-v13-n5-p02-s.pdf claim: Theorem 1.4 gives an unconditional effective relative-error prime count at polynomial conductor scale; its quadratic inert class supplies negative-character primes in a terminal dyadic interval with ramified zeros retained. strata_touched: [] license: citation-only triage: anchor

Quadratic inert-prime supply at polynomial conductor scale

The published article is Algebra & Number Theory 13 (2019), 1039–1068, DOI:10.2140/ant.2019.13.1039. The inspected primary statements are Theorem 1.1, printed p.1040, and Theorem 1.4 with parameter (1-11), printed p.1042, in the journal PDF. The corresponding preprint is arXiv:1803.02823v3. This note applies the published theorem; it does not reproduce its analytic proof or supply Lean verification, numerical onset constants, or a claim that this is the strongest available interval theorem.

Let $\chi$ be a real nonprincipal primitive quadratic character of conductor $q\ge 3$, corresponding to the quadratic field $L=\mathbb{Q}(\sqrt{\Delta })$ with signed fundamental discriminant $|\Delta |=q$. In Theorem 1.4 take $F=K=\mathbb{Q}$, $H=G=\mathrm{Gal}(L/\mathbb{Q})$, and the nontrivial singleton Frobenius class. Its parameters are $D_K=n_K=1$ and $Q(L/K)=q$. The count is precisely

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

Ramified primes remain zeros and are absent from this count. The theorem supplies absolute effective constants $a,c>0$ and an absolute effective implied constant such that, for $x\ge q^a$,

$${\pi }_{-}(x)=M(x)\left(1+O\left(e^{-c\log x/\log q}+e^{-\sqrt{c\log x}}\right)\right).\text{\hspace{1em}(T1)}$$

Here $M(x)=\mathrm{Li}(x)/2$ without an exceptional zero, and $M(x)=[\mathrm{Li}(x)+\mathrm{Li}(x^{{\beta }_1})]/2$ if one exists. Since $\zeta (s)$ has no real zero in $(0,1)$, the exceptional character in this quadratic extension is the nontrivial character. Thus Theorem 1.4 has ${\theta }_1=-1$ on the inert class, giving the plus sign. No RH or GRH hypothesis is used.

The interval consequence needed by an actual CA source

There are absolute effective constants $C\ge 1$ and $x_0$ such that every such character and every $x\ge \max \{x_0,q^C\}$ satisfy

$${\pi }_{-}(x)-{\pi }_{-}(x/2)\ge \frac{x}{8\log x}.\text{\hspace{1em}(T2)}$$

This is a consequence of (T1), not a separately quoted source theorem. Enlarge $C,x_0$ until (T1) applies at both endpoints and each relative error is at most $1/64$. Uniformity follows from $\log (x/2)/\log q\ge C-\log 2/\log 3$. For sufficiently large $x$, $\mathrm{Li}(t)\le 2t/\log t$ at $t=x,x/2$, so $M(x)+M(x/2)\le 4x/\log x$. Also

$$M(x)-M(x/2)\ge \frac{1}{2}{\int }_{x/2}^x\frac{dt}{\log t}\ge \frac{x}{4\log x}.$$

The exceptional term is increasing and cannot reduce this difference. Subtracting the two endpoint errors gives at least $(1/4-4/64)x/\log x\ge x/(8\log x)$, proving (T2). The constants are effective existence constants, not computed numerical thresholds. A least nonresidue below $q^b$, or a prime count accumulated below that cutoff, would not alone supply this interval conclusion.

The actual CA application uses (T2) only where the same candidate’s terminal support has exponent one. No large-conductor weighted deficit or signed Robin estimate is supplied by this note.