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Statement

Theorem 1.5 (p. 2, quoted). "Let $c_2=1/(4\pi)$. If the Riemann hypothesis is true, then $\pi(m+n)\le\pi(m)+\pi(n)$ for all integers $m\ge n\ge2$ satisfying $n\ge c_2\sqrt m\log m\log(m\log^8m)$."

The theorem is conditional on the Riemann hypothesis.

Proof pointer

Section 7, pp. 7--8. The input is Dusart's Proposition 7.1 (p. 7): under the Riemann hypothesis, $|\pi(x)-\operatorname{li}(x)|\le\frac{\sqrt x}{8\pi}\log\frac{x}{\log x}$ for real $x\ge5639$. For $m\le5\times10^{19}$ one has $m+n\le10^{20}$ and the result follows from Theorem 1.4. For larger $m$, pairs with $n\ge c_0m/\log^2m$ are covered by Theorem 1.3; otherwise the mean value theorem, Proposition 7.1 and Dusart's bound $\pi(t)\ge t/(\log t-1)$ for $t\ge5393$ give the reduction (7.1), which the paper checks in three ranges of $n$, (7.2) to (7.4).

Read depth

Claims checked: the statement and Proposition 7.1 were read clause by clause on the pages of the copy named on the source card. The proof was read but not checked. Nothing here is independently reviewed.

Dependencies

• The Riemann hypothesis, as a hypothesis.
• Proposition 7.1 (p. 7), cited from P. Dusart, Ramanujan J. 47 (2018), 141--154, Proposition 2.6.
• Theorem 1.3 and Theorem 1.4.

Source. Christian Axler, "Some Results on a Conjecture of Hardy and Littlewood," arXiv:1909.12625v2 (2019), the edition read for the source card.

Bears on

• Problem 855: assuming the Riemann hypothesis, the theorem proves the problem's inequality for integers $X\ge Y\ge2$ with $Y\ge c_2\sqrt X\log X\log(X\log^8X)$. Pairs with smaller $Y$ remain uncovered even under that hypothesis, so it does not decide the problem.