On the Second Hardy-Littlewood Conjecture
Bittu Chahal, Ertan Elma, Nic Fellini, Akshaa Vatwani, Do Nhat Tan Vo
TL;DR
The paper investigates the second Hardy–Littlewood conjecture, which asks whether the prime counting function satisfies the subadditivity inequality $π(x+y)≤π(x)+π(y)$. By linking the subadditivity to the error term in the Prime Number Theorem through a function $R(x)$ with $|π(x)-\mathrm{li}(x)|≤C R(x)$, the authors derive unconditional improvements on the range of $y$ for which subadditivity holds and establish RH-based ranges with explicit lower bounds in terms of $x$. Under RH, they prove that for any $ε>0$, there exists $x_ε$ such that for all $x≥x_ε$ and $y$ in $\frac{(2+ε)\sqrt{x}\log^2 x}{8π}≤y≤x$, one has $π(x+y)≤π(x)+π(y)$, with further refinements giving precise $o(1)$-type corrections. They also provide corollaries on conditional bounds assuming RH up to a height and quantify the size of the exceptional set of pairs $(x,y)$ where the inequality fails, including extensions to primes in arithmetic progressions. These results advance understanding of when the second Hardy–Littlewood conjecture holds and quantify how often it may fail depending on the error term in the PNT.
Abstract
The second Hardy-Littlewood conjecture asserts that the prime counting function $π(x)$ satisfies the subadditive inequality \begin{align*} π(x+y)\leqslant π(x)+π(y) \end{align*} for all integers $x,y\geqslant 2$. By linking the subadditivity of $π(x)$ to the error term in the Prime Number Theorem, we obtain unconditional improvements on the range of $y$ for which $π(x)$ is known to be subadditive. Moreover, assuming the Riemann Hypothesis, we show that for all $ε>0$, there exists $x_ε \geqslant 2$ such that for all $x\geqslant x_ε$ and $y$ in the range \begin{align*} \frac{(2+ε)\sqrt{x}\log^2x}{8π}\leqslant y\leqslant x, \end{align*} the inequality $π(x+y)\leqslant π(x) + π(y)$ holds.