Suppose $f:\mathbb{Z}_{\geq 0}\to\mathbb{Z}_{\geq 0}$ is an arithmetic function that grows slower than the identity map. Has it already been conjectured that, under this general hypotheses, $\pi(n+f(n))\pi(nf(n))\sim \dfrac{2f(n)}{\log n}$? Or are there any known counterexamples?
Thanks in advance.



This paper by Yildirim gives a good survey on this. Clearly, the question is equivalent to determining functions $\Phi$ such that $\pi(x+\Phi(x))\pi(x)\sim \frac{\Phi(x)}{\log x}$. Quoting the paper, this was proved by HeathBrown for $\Phi(x)=x^{\frac{7}{12}\varepsilon(x)},\varepsilon(x)\to 0$, and assuming the Riemann hypothesis, we can take $\Phi(x)=x^{\frac{1}{2}+\varepsilon}$. However, $\Phi$ cannot increase too slowly, as $\Phi(x)=(\log x)^{\lambda}$ does not work for any $\lambda$ due to a result of Maier. 

