On the second Hardy–Littlewood conjecture
Volume 223 / 2026
Acta Arithmetica 223 (2026), 185-196
MSC: Primary 11N05
DOI: 10.4064/aa250808-4-11
Published online: 1 April 2026
Abstract
The second Hardy–Littlewood conjecture asserts that the prime counting function $\pi (x)$ satisfies the subadditive inequality $$\pi (x+y)\leqslant \pi (x)+\pi (y)$$ for all integers $x,y\geqslant 2$. By linking the subadditivity of $\pi (x)$ to the error term in the Prime Number Theorem, we obtain unconditional improvements on the range of $y$ for which $\pi (x)$ is known to be subadditive. Moreover, assuming the Riemann Hypothesis, we show that for all $\epsilon \gt 0$, there exists $x_{\epsilon} \geqslant 2$ such that for all $x\geqslant x_\epsilon $ and $y$ in the range $$ \frac{(2+\epsilon )\sqrt{x}\log ^2x}{8\pi}\leqslant y\leqslant x, $$ the inequality $\pi (x+y)\leqslant \pi (x) + \pi (y)$ holds.
