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Source. Lemma 4.1 (pp. 9--10) and Theorem 4.2 (p. 10), Section 4, of Nathan McNew, The convex hull of the prime number graph, in: Irregularities in the Distribution of Prime Numbers, Springer, Cham (2018), 125--141, doi:10.1007/978-3-319-92777-0_7, cited at the page numbers 1--15 of the author's preprint named on the source card.

Statement

The log-prime number graph is the set of points $(n,\log p_n)$, and the log-convex primes are the primes $p_n$ whose point is a vertex of its convex hull (pp. 2, 9). The paper relates them (p. 2) to the good primes, those with $p_n^2>p_{n-i}p_{n+i}$ for all positive $i<n$. Here $\mathrm{ali}$ is the inverse function of $\mathrm{li}$ (p. 9).

Lemma 4.1 (pp. 9--10). If $(m,\log p_m)$ is any point on the boundary of the convex hull of the log-prime number graph, the segment of the hull following it has slope

$$\frac{\mathrm{ali}'(m)}{\mathrm{ali}(m)}+O\Bigl(\frac1m\exp\Bigl\{-\frac{A\log^{3/5}m}{2(\log\log m)^{1/5}}\Bigr\}\Bigr)$$

as $m\to\infty$, for some $A>0$.

Theorem 4.2 (p. 10). As $x\to\infty$, the number of log-convex primes up to $x$ is at most

$$\frac{x}{\log^{4/3-o(1)}x}.$$

So the log-convex primes have relative density zero among the primes, which Pomerance had conjectured (p. 9). The bound is for the log-convex primes only; whether the good primes are $o(\pi(x))$ is left open as Question 5.1 (p. 14).

Read depth. Claims checked: Lemma 4.1 and Theorem 4.2 were read clause by clause on the page images of the preprint; the proofs were read but not checked, and nothing here is independently reviewed.

Proof pointer

pp. 10--11. Take consecutive log-convex primes $p_n<p_{n+k}$ with $x/\log x\le p_n<p_{n+k}\le x$ and $k\le\log^{1/3}x$. Assuming $p_{n+k}-p_n\le n^{\epsilon}$, Lemma 4.1 gives $|(p_{n+k}-p_n)-k\log x|\le kb\log\log x$ for an absolute constant $b$. A Brun sieve bound for prime pairs with a fixed difference, summed with Mertens's theorem over these differences and over $k$, bounds the number of such pairs by a constant times $(\log\log x)^2x/\log^{4/3}x$. Log-convex primes not in such a pair are at least $\log^{1/3}x$ primes apart, so they number less than $\pi(x)/\log^{1/3}x$.

Dependencies

Lemma 4.1 (above), the prime number theorem with error term (equation (15), p. 10), Brun's sieve and Mertens's theorem.

Bears on

No Erdős problem directly.