Smallest $\epsilon > 0$ for which there's always a prime between $n$ and $(1+\epsilon)n$

Question

One can show via the PNT that $$\lim_{n\to \infty} \frac{\pi((1+\epsilon)n) - \pi(n)}{n/\log{n}} = \epsilon,$$

for any $\epsilon > 0$, which in particular implies that for any $\epsilon > 0$ and for any $n$ big enough there is always a prime between $n$ and $(1+\epsilon)n$.

Bertrand's postulate is a special case of this $(\epsilon = 1)$ with the added benefit that it holds for all $n > 1$, so it works for all positive integers (excluding 1).

What's the smallest known $\epsilon > 0$ such that there is always a prime between $n$ and $(1+\epsilon)n$? Clearly we must have $\epsilon > 1/2$ so that $(1+\epsilon)2 > 3$, but i'm not sure $1/2$ works.

Answer 1

If you want it to work for all $n\geq 2$ then let $\epsilon_n = \frac{p_{n+1}}{p_n}-1$ and take the maximum $n$, from PNT we know that $ \lim \limits_{n \to \infty}\frac{p_{n+1}}{p_n}=1 $ and so $ \lim \limits_{n\to \infty} \epsilon_n =0$ and checking for smaller cases we find that $\epsilon_2 = \frac{5}{3}-1 \approx 0.666$ is the minimum epsilon that the interval $ [n ,n(1+\epsilon)]$ is garneted to contains a prime

Answer 2

By the most recent development we are not looking into the constant any longer. It seems that the latest constant that is investigated is $\frac{1}{16597}$. It turned out that $\epsilon \to 0$ as we approach infinity. So the best known results in this format are all in the form

$$x < p \leq (1+\frac1{a\log^b(x)})x, x > M$$

where as $M$ grows the $\epsilon$ is all smaller.

For example $(M=89693,a=1,b=3)$ or $(M=468991632,a=5000,b=2)$.

Here, we can argue if this is really the best way of describing the gap between primes because the function in question $\log(x)$ runs slowly and then there will be always some better expression for a large portion of primes while asymptotically that is not the best result.

At the moment all gaps that we can conjecture and experimental results are far apart (it is very typical that we cannot prove that there is one prime number, while numerical evidence is showing that there are thousands) to the extent that we have to say that we do not know much about what gap dynamics really looks like. Some result are now almost a century old and very difficult to improve even by one decimal notch.