Commonly known as the second Hardy-Littlewood conjecture. In [Er85c] Erdős describes it as 'an old conjecture of mine which was probably already stated by Hardy and Littlewood'.

This is probably false, since Hensley and Richards [HeRi73] have shown that this is false assuming the Hardy-Littlewood prime tuples conjecture. Indeed, assuming this conjecture, for every large $y$ there are infinitely many $x$ such that\[\pi(x+y)>\pi(x)+\pi(y)+c\frac{y}{\log y}\]for some absolute constant $c>0$.

Erdős [Er85c] reports Straus as remarking that the 'correct way' of stating this conjecture would have been\[\pi(x+y) \leq \pi(x)+2\pi(y/2).\]Clark and Jarvis [ClJa01] have shown this is also incompatible with the prime tuples conjecture.

In [Er85c] Erdős conjectures the weaker result (which in particular follows from the conjecture of Straus) that\[\pi(x+y) \leq \pi(x)+\pi(y)+O\left(\frac{y}{(\log y)^2}\right),\]which the Hensley and Richards result shows (conditionally) would be best possible. Richards conjectured that this is false.

Erdős and Richards further conjectured that the original inequality is true almost always - that is, the set of $x$ such that $\pi(x+y)\leq \pi(x)+\pi(y)$ for all $y<x$ has density $1$. They could only prove that this set has positive lower density.

They also conjectured that for every $x$ the inequality $\pi(x+y)\leq \pi(x)+\pi(y)$ is true provided $y \gg (\log x)^C$ for some large constant $C>0$.

Hardy and Littlewood proved\[\pi(x+y) \leq \pi(x)+O(\pi(y)).\]The best known in this direction is a result of Montgomery and Vaughan [MoVa73], which shows\[\pi(x+y) \leq \pi(x)+2\frac{y}{\log y}.\]This is discussed in problem A9 of Guy's collection [Gu04].

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This page was last edited 28 September 2025.

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