On the Second Hardy-Littlewood Conjecture

The second Hardy-Littlewood conjecture asserts that the prime counting function $\pi (x)$ satisfies the subadditive inequality \begin{align*} π(x+y)\leqslant π(x)+π(y) \end{align*} for all integers $x,y⩾2$. 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 $\epsilon >0$, there exists $x_{\epsilon }⩾2$ such that for all $x⩾x_{\epsilon }$ and $y$ in the range \begin{align*} \frac{(2+ε)\sqrt{x}\log^2x}{8π}\leqslant y\leqslant x, \end{align*} the inequality $\pi (x+y)⩽\pi (x)+\pi (y)$ holds.