On the Second Hardy-Littlewood Conjecture

Abstract

The second Hardy-Littlewood conjecture asserts that the prime counting function $\ddot{I}€(x)$ 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 $\ddot{I}€(x)$ to the error term in the Prime Number Theorem, we obtain unconditional improvements on the range of $y$ for which $\ddot{I}€(x)$ is known to be subadditive. Moreover, assuming the Riemann Hypothesis, we show that for all $\hat{I}\mu >0$, there exists $x_{\hat{I}}\mu ⩾2$ such that for all $x⩾x_{\hat{I}}\mu$ and $y$ in the range \begin{align*} \frac{(2+ε)\sqrt{x}\log^2x}{8π}\leqslant y\leqslant x, \end{align*} the inequality $\ddot{I}€(x+y)⩽\ddot{I}€(x)+\ddot{I}€(y)$ holds.

10 pages