bibkey: lichtmanmartinpomerance2019primeraces authors: Jared Duker Lichtman; Greg Martin; Carl Pomerance year: 2019 title: Primes in prime number races doi: 10.1090/proc/14569 url: https://arxiv.org/abs/1809.03033v2 claim: Theorem 2.1 transfers fixed-level logarithmic densities to prime samples under explicit stability and distribution hypotheses; the Mertens application assumes RH and LI and supplies no finite growing-level Robin count. strata_touched: [] license: citation-only triage: anchor

Fixed-level densities and actual prime samples

The inspected primary is the versioned HTML of arXiv:1809.03033v2, revised 5 January 2019. The abstract metadata lists Proceedings of the American Mathematical Society 147 (2019), 3743–3757 and the DOI above. The journal edition and complete proof were not independently audited. All theorem locators below refer to the inspected manuscript. This note cites the results and does not rerun the paper’s prime computations or claim Lean certification.

The two hypotheses of the general sampling theorem

For $f:[1,\infty )\to \mathbb{R}$ and fixed $a\in \mathbb{R}$, put ${\mathcal{M}}_a(f)=\{x:f(x)>a\}$. Theorem 2.1, §2, assumes both:

• For each pair of fixed real levels $a>b$, there is $x_0(a,b)$ such that $x\ge x_0$ and $f(x)>a$ imply $f(z)>b$ for every $z\in [x,x+x^{1/3}]$. The same condition holds for $-f$.
• Every ${\mathcal{M}}_a(f)$ has logarithmic density, and $a\mapsto \delta ({\mathcal{M}}_a(f))$ is continuous.

Then, for each fixed $a$, the relative prime density exists and equals the continuous logarithmic density:

$$\underset{Y\to \infty }{\lim }\frac{1}{\log Y}\sum _{\begin{array}{c}p\le Y\\ f(p)>a\end{array}}\frac{\log p}{p}=\underset{Y\to \infty }{\lim }\frac{1}{\log Y}{\int }_{\begin{array}{c}1\le x\le Y\\ f(x)>a\end{array}}\frac{dx}{x}.$$

This is the source’s ${\delta }^{\ast }$, not an unweighted finite interval count. Section 1 distinguishes it from the $1/p$-weighted density normalized by $\log \log Y$: existence of ${\delta }^{\ast }$ implies equality of those limits, but the converse is not asserted. The general theorem has no RH hypothesis; its two hypotheses must still be proved for any proposed input function.

The Mertens application keeps RH and LI

Section 4 uses the strict prime-product endpoint

$$E_M(x)=\sqrt{x}\log x\left[\log \prod _{q<x}(1-1/q{)}^{-1}-\log \log x-\gamma \right].$$

Theorem 4.1 assumes RH and the paper’s linear independence hypothesis for positive zeta-zero ordinates. It gives relative logarithmic density $1-\Delta$ for the primes with $E_M(p)>0$, where $\Delta$ is the Rubinstein–Sarnak prime-race density recalled in equation (1.2). The local stability and conditional limiting distribution used there are existing arguments, not new project proofs.

The missing finite-scale and actual-source comparison

The actual CA counting application needs an unweighted count in a dyadic prime range at a threshold $\delta X^{-\eta }$, with a particular fixed $0<\eta <1/2$ supplied by the selected source. On the $\sqrt{x}\log x$ normalization, this is a growing level of order $X^{1/2-\eta }\log X$, rather than a fixed $a$. The theorem supplies neither a convergence rate nor a uniform growing-level estimate in this range.

The project’s Nicolas surplus uses a weak prime-product endpoint and denominator $\log \vartheta (p)$, while $E_M$ uses $q<p$ and denominator $\log p$. The complete post-event Robin quantity also retains every exponent numerator and the actual size $\log C_{{\tau }_p}$. These are distinct readouts. A finite joint comparison must pay all of their differences before transporting a density or count.

Thus the general theorem is a reusable sampling result under its stated hypotheses; its RH-and-LI Mertens application is not an unconditional supplier for the selected Robin source. No signed finite-scale count, full Robin tail estimate or RH proof is obtained.