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Claim. C. Axler, Some results on a conjecture of Hardy and Littlewood, arXiv:1909.12625 (first version 27 September 2019; the second, of 30 September 2019, is the copy read), Theorem 1.3 (p. 2): with $c_0=0.70881678090424862707121$,

$$\pi(m+n)\le\pi(m)+\pi(n)$$

for all integers $m\ge n\ge2$ with $n\ge c_0m/\log^2m$. The proof (pp. 5--6) applies Proposition 5.1, which turns explicit two-term bounds for $\pi$ into a range of the inequality, together with the computer check of Proposition 2.4 (p. 2), which gives the inequality whenever $m+n\le39\,708\,229\,123$. Theorem 1.1 (p. 2) proves the cone $m/1950\le n\le m$, and Theorem 1.5 (p. 2) the range $n\ge\sqrt m\log m\log(m\log^8m)/(4\pi)$ under the Riemann hypothesis; neither widens the unconditional range for large pairs. The paper is compiled at its card, whose statements were read clause by clause and whose proofs were read but not checked.

Covers. Every integer pair whose smaller entry $n\ge2$ and larger entry $m$ satisfy $n\ge c_0m/\log^2m$. For large pairs this contains the ranges of Dusart's claim and Alkan's claim. Not covered: the pairs with $n<c_0m/\log^2m$, which include pairs with both entries arbitrarily large, so the problem as posed is not settled.

Depends on. Nothing in this wiki.

Standing. The paper is on arXiv only: its record carries no journal reference, and no published version is known. So the claim is claimed with no evidence listed. The page is named by the first version's posting date.