Does Bertrand's Postulate give us the tightest proven upper bound for prime gaps?

Bertrand's Postulate asserts that there is a prime between $n$ and $2n$.

Is this the best such upper bound on prime gaps known today, or have stronger estimates been proved? I mean results of the kind:

• there will always be a prime between $n$ and $2n-2$, or
• there will always be a prime between $n$ and $cn$ with $1<c<2$?

Was any such improvement proved rigorously, or is Bertrand's Postulate still the best we have?

• You can read Wikipedia. There is no reason to ask such a question here. – Marc van Leeuwen Sep 27 '14 at 4:39
• There is a nice proof of this from the book "Proofs from the book", which is elementary and you might like. – Bombyx mori Sep 27 '14 at 5:11

Yes. See theorem 3 of:

O. Ramar´e, Y. Saouter, Short effective intervals containing primes, J. Number Theory 98 (2003), no. 1.

A related statement states for all $\epsilon > 0$, for sufficiently large $n$, there is a prime between $n, (1+ \epsilon)n$ which is a trivial corollary of the PNT.

For your first question, both of those statements have been proven.

• "sufficiently large" makes this a non-generalisation of Bertrands postulate. – Marc van Leeuwen Sep 27 '14 at 4:42
• What about my second question? All known primes gaps have been observed to be smaller than n or n-2, with maybe the exception of the largest known prime. – Jeffrey Young Sep 27 '14 at 5:10
• Have you looked at the reference? And what do you mean by "with maybe the exception of the largest known prime"? – Mayank Pandey Sep 27 '14 at 5:27
• @MayankPandey From the largest known prime to the next prime, what is the distance? Let k be the largest known prime. Let u be the next consecutive prime, which is currently unknown. Will u be only slightly less than 2k? – Jeffrey Young Sep 27 '14 at 6:28
• I should have said that all known prime gaps are significantly smaller than that which is allowed by Bertrand's postulate. However, we do not know concerning any prime gaps following the largest known prime as to whether they will be as large as Bertrand's postulate will allow or if Legendre's conjecture will hold true. – Jeffrey Young Sep 27 '14 at 6:49