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Statement

Theorem 1.3 (p. 2, quoted). "Let $c_0=0.70881678090424862707121$. Then we have $\pi(m+n)\le\pi(m)+\pi(n)$ for all integers $m\ge n\ge2$ with $n\ge c_0m/\log^2m$."

The paper presents it (p. 2) as a refinement of Panaitopol's range $\pi(m)\le n\le m$, display (1.6).

Proof pointer

Section 5, pp. 5--6. The theorem is Proposition 5.1 with $k=2$, $a_1=1$, $a_2=2.85$, $\alpha_2=38\,099\,531$, $\varepsilon=0.70863503301170907614119$, $\beta_2=14\,000\,264\,036\,190\,262$, $\gamma_2=23$ and $c=c_0$, the values of $\alpha_2$ and $\beta_2$ taken from Axler's earlier paper (its reference [2], Corollary 3 and Theorem 2). This gives the inequality for $m\ge14\,000\,264\,036\,190\,263$ and $n\ge c_0m/\log^2m$. For smaller $m$ one has $c_0/\log^2m\ge1/1950$, and the claim follows from Theorem 1.1. The printed proof (p. 6) says the values are substituted "into Proposition 3.1" [sic]; the substitution described is into Proposition 5.1.

Read depth

Claims checked: the statement, Proposition 5.1 and the parameter values were read clause by clause on the pages of the copy named on the source card. The proof was read but not checked, and the constants were not recomputed. Nothing here is independently reviewed.

Dependencies

• Proposition 5.1 (p. 5).
• Theorem 1.1 (p. 2), for $m\le14\,000\,264\,036\,190\,262$.
• Explicit bounds for $\pi(x)$ from C. Axler, Integers 18 (2018), Paper No. A52 (see its card).

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: with integers $X\ge Y\ge2$ the two arguments, the theorem proves the problem's inequality whenever $Y\ge c_0X/\log^2X$, so any pair of integers violating it has $Y<c_0X/\log^2X$. It says nothing about that remaining region and does not decide the problem.