What is the smallest known $c$ so that for any $n\geq 2$ there are at least $n/\log\_2{n}$ primes between $n$ and $cn$ (inclusive)?


The [prime number theorem](http://en.wikipedia.org/wiki/Prime_number_theorem) seems to give an asymptotic result so I am not sure how to get a particular constant from it.


I am interested in answers that depend on well known conjectures as well as answers that are known to be correct.


Following a comment, the notation $\log\_2{n}$ refers to taking the logarithm base $2$.


[Cross-posted from [math.se](https://math.stackexchange.com/questions/1281888/the-number-of-primes-in-an-interval) where $c=13/6$ was suggested as a possibility.]



Smallest constant so that there are at least $n/\log_2{n}$ primes between $n$ and a constant multiple of $n$
Assuming I made no mistake, $c=11/5$ is the smallest such value.


Let $c=11/5$. Using Theorem 1 of [Rosser-Schoenfeld](https://projecteuclid.org/euclid.ijm/1255631807), we see easily that
$$\pi(cn)-\pi(n-1)\geq(\log 2)\frac{n}{\log n},\qquad n\geq 74,$$
where $\log$ is the natural logarithm as usual in analytic number theory. We can verify the same inequality for $2\leq n\leq 73$ by computer. Finally, for $n=5$ the above inequality means that there are at least $3$ primes in $[5,5c]$, whence $c\geq 11/5$ is necessary.


