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Claim. E. Alkan, A generalization of the Hardy-Littlewood conjecture, Integers 22 (2022), #A53, Theorem 1 (p. 4): if a set $P$ of primes has counting function $\pi_P(x)=c\pi(x)+O(x/(\log x)^3)$ for a constant $0<c\le1$, then there are effective constants $K,x_0>0$, depending on $P$, such that

$$\pi_P(x+y)<\pi_P(x)+\pi_P(y)$$

for all $x,y\ge x_0$ with $Kx/\log x\le y\le x$. With $P$ the set of all primes the hypothesis holds with $c=1$, and the theorem gives the problem's inequality, strictly, on that region. Corollary 1 (pp. 4--5) deduces that at most $O(X^2/\log X)$ pairs with entries at most $X$ fail it. The paper is compiled at its card, which holds the published article. Its conditional violations under the prime $k$-tuples conjecture, display (4.5), rest on the estimate recorded on Hensley and Richards's claim page.

Covers. Every pair with both entries at least $x_0$ whose smaller entry $u$ and larger entry $v$ satisfy $u\ge Kv/\log v$. For large $v$ this region lies inside the range of Dusart's claim, so the claim adds the strict inequality there and no new pairs. Not covered: the pairs with $u<Kv/\log v$, which include pairs with both entries arbitrarily large, so the problem as posed is not settled.

Depends on. Nothing in this wiki.

Acceptance. Refereed: Integers 22 (2022), a refereed journal (received 3 February 2022, accepted 27 May 2022, published 3 June 2022, as the article prints). The page carries the publication date.