Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$.
What is
$$
 \limsup\_{m \rightarrow \infty} \left( \limsup\_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} \right)?
$$
If needed, answers may be conditional under the assumption that a suitable generalization
of the [Bunyakovsky conjecture](https://en.wikipedia.org/wiki/Bunyakovsky_conjecture) holds.



How many primes can there be in a short interval?
As observed by Hensley and Richards in


*Douglas Hensley and Ian Richards*, [**Primes in intervals**](http://www.ams.org/mathscinet-getitem?mr=396440), *Acta Arith.* **25** (1973-74), 375--391, 


if the prime tuples conjecture is true, then $\limsup\_{n \to \infty} \pi(n+m) - \pi(n) = \rho^\*(m)$, where $\rho^\*(m)$ is the length $k$ of the largest admissible $k$-tuple of diameter less than $m$ (a $k$-tuple is admissible if it avoids a residue class modulo $p$ for every prime $p$). So your quantity would then be equal to $\limsup\_{m \to \infty} \rho^\*(m)/\pi(m)$, which is also easily seen to be equal to $\limsup\_{k \to\infty} k \log k / H(k)$, where $H(k)$ is the minimal diameter of an admissible $k$-tuple.


The best known upper and lower bounds for $H(k)$ can be found in Section 3 of [the (unabridged version of) the Polymath8a paper](http://arxiv.org/pdf/1402.0811v2.pdf). The best asymptotic upper bound known is $k \log k + k \log\log k - (1+2 \log 2) k + o(k)$ (due to Schinzel), and the best asymptotic lower bound is a bit tricky to state (coming from the sharp Montgomery-Vaughan version of the large sieve) but is roughly of the form $\frac{1}{2} k \log k - o(k \log k )$. So the best that is known about your quantity is (as was pointed out in comments) it is somewhere between $1$ and $2$. (One does not need the best results on $H(k)$ for these bounds: the lower bound comes from observing that the first $k$ primes larger than $k$ are automatically admissible, while the upper bound follows from (the proof of) the Brun-Titchmarsh inequality.) Based on the known numerical values and bounds for $H(k)$ (see [this online database](http://math.mit.edu/~primegaps/), as well as Figures 2 and 3 of the Polymath8a paper referenced above) one can tentatively conjecture that your quantity is equal to $1$, but the evidence is not terribly strong for this conjecture yet. Improving the upper bound of $2$ would be a major breakthrough in sieve theory (closely related to the parity problem), and is one of the motivations of the nascent field of "inverse sieve theory", discussed in


*Ben Green and Adam J. Harper*, [**Inverse questions for the large sieve**](http://dx.doi.org/10.1007/s00039-014-0288-1), *Geom. Funct. Anal.* **24** (2014), no. 4, 1167--1203.


The upper bound of $2$ is unconditional (following directly from [Brun-Titchmarsh](https://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem)). No non-trivial unconditional lower bound (beyond the trivial bound of $0$) is known; the best result is by Maynard in


*James Maynard*, [**Small gaps between primes**](http://dx.doi.org/10.4007/annals.2015.181.1.7), *Ann. of Math. (2)* **181** (2015), no. 1, 383--413 


with slightly better implied constants obtained by the Polymath8b project


*D. H. J. Polymath*, [**Variants of the Selberg sieve, and bounded intervals containing many primes**](http://dx.doi.org/10.1186/s40687-014-0012-7), *Res. Math. Sci.* **1** (2014), Art. 12, 83,


basically showing that $\limsup\_{n \to \infty} \pi(n+m) - \pi(n) \gg \log m$. This is known to be the limit of the existing Selberg sieve method to unconditionally produce many primes in short intervals; getting beyond the $\log m$ bound here would be a significant breakthrough (though I think breaking the parity upper bound of $2$ mentioned earlier would be even more revolutionary).


Note that until the famous recent breakthrough of Zhang in


*Yitang Zhang*, [**Bounded gaps between primes**](http://dx.doi.org/10.4007/annals.2014.179.3.7), *Ann. of Math. (2)* **179** (2014), no. 3, 1121--1174.


it was not even known unconditionally that $\limsup\_{n \to \infty} \pi(n+m) - \pi(n)$ was larger than $1$ for any $m$.


EDIT: Ben Green pointed out to me an observation of Selberg in Vol. I of his "Collected works" (page 610 in the new edition) that one cannot obtain effective improvements to the $1/2$ constant in the lower bound $H(k) \geq \frac{1}{2} k \log k - o(k \log k)$ without also getting effective bounds in Siegel's theorem, since a Siegel zero implies the existence of a medium-sized arithmetic progression with about twice as many primes as one would expect without the zero. This leads to an admissible k-tuple that is also about twice as dense as expected. So the upper bound of 2 in the original question is unlikely to be improved without some sort of breakthrough on the Siegel zero problem (or else some massive extension of the repulsion phenomenon).


