Problem 855

Erdős Problem #855: declared status 'open'. Formalized: yes. If $\pi(x)$ counts the number of primes in $[1,x]$ then is it true that (for large $x$ and $y$)$$\pi(x+y) \leq \pi(x)+\pi(y)?$$ Current best: 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]. Prize: no. OEIS: A023193. Tags: number theory, primes.