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Claim. B. Chahal, E. Elma, N. Fellini, A. Vatwani and D. N. T. Vo, On the second Hardy--Littlewood conjecture, arXiv:2503.02766v1 (4 March 2025), Theorem 1.1 (p. 3): let $C>0$, $x_0\ge2$ and a positive nondecreasing function $R$ satisfy $|\pi(x)-\operatorname{li}(x)|\le CR(x)$ for $x\ge x_0$ (display (1.4)); then for $x\ge x_0$ and

$$\frac{3CR(2x)\log^2x}{\log\log x}\le y\le x$$

$\pi(x+y)\le\pi(x)+\pi(y)$. Any unconditional error term of the prime number theorem may serve as $R$; the paper lists explicit ones, (1.5)--(1.7), cited from Johnston and Yang and from Mossinghoff, Trudgian and Yang, and with them the lower end of the range is $x$ times a factor that tends to zero faster than any power of $1/\log x$ (Corollary 1.5, p. 4, gives the matching count of possible exceptions). The proof (Section 2.1) compares $\operatorname{li}(x)+\operatorname{li}(y)-\operatorname{li}(x+y) \ge y\log\log x/\log^2x$ with the three error terms. Under the Riemann hypothesis, Corollary 1.2 (p. 3) gives the inequality for $x\ge x_\epsilon$ and $(2+\epsilon)\sqrt x\log^2x/(8\pi)\le y\le x$; that conditional range is recorded here and is not part of this page's scope. The paper is compiled at its card, whose statements were read clause by clause on arXiv v1 with the proofs of Section 2 followed.

Covers. Every pair whose larger entry $x\ge x_0$ and smaller entry $y$ satisfy $y\ge3CR(2x)\log^2x/\log\log x$ for an unconditional error term $R$. For large pairs this contains the ranges of Dusart's claim, Alkan's claim and Axler's claim. Not covered: smaller $y$, for instance $y=\lfloor\sqrt x\rfloor$, so the problem as posed is not settled.

Depends on. Nothing in this wiki.

Acceptance. Refereed: Acta Arithmetica 223 (2026), 185--196, doi:10.4064/aa250808-4-11, published online 1 April 2026 (the publisher's article page and the Crossref record). The journal's abstract states the unconditional improvement of the range and the Riemann-hypothesis range of Corollary 1.2. The statements above are read from arXiv v1; the journal text was not compared. The page is named by the first version's posting date.