TITLE: Is there a progress on a solution of the inequality $\pi (m+n) \leq \pi (m) + \pi (n)$
QUESTION [17 upvotes]: in 1923 Hardy and Littlewood proposed the conjecture $\pi (m+n) \leq \pi (m) + \pi (n)$. Is there any progress towards solving this conjecture?

REPLY [7 votes]: As has already been pointed out, Montgomery and Vaughn have proven that
$$\pi(x+y) \leq \pi(x)+ 2 \frac{y}{\log(y)} $$
from which, with some care, one can derive that: $\pi(x+y)  \leq \pi(x)+ 2\pi(y)$ (this is worked out in the original paper of Montgomery and Vaughn). 
The natural question is thus to further refine the constant $2$. There seems to be no unrestricted result on this problem, however Friedlander and Iwaniec (in their recent book on sieve theory) have shown that
$$\pi(x+y) \leq  \pi(x) + (2-\delta) \frac{y} {\log(y)} $$
holds for $x^{\theta} < y < x$ where $\delta:= \delta(\theta)$ is a function of $\theta$. In other words, one can improve the constant $2$ as long as $y$ isn't too small compared to $x$.  
Recently Bourgain and Garaev have refined this result to give the quantitative relationship of $\delta \sim \theta^2$.