The Riemann hypothesis implies that $d_n=p_{n+1}-p_n=O(n^{1/2}\log n)$. Is there a bound on $d_n$ that would imply the truth of the Riemann hypothesis?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
No, this is more or less unrelated. It is generally believed that $d_n=O((\log n)^2)$ (and more precise conjectures exist; see Cramér–Granville Conjecture), so much better than what RH is known to give, but this is somehow independent of the Riemann Hypothesis. For example, see Granville - Harald Cramér and the distribution of prime numbers around gaps between primes, with no mention of anything like this.
Also note that the results on $d_n$ one can obtain uncoditionally are in some sense not that far away of what one gets under Riemann Hypothesis, namely Baker–Harmann–Pintz have the result you mention with $0.525$ instead of $0.5$ (essentially, i.e., ignoring logarithmic terms).
This can be contrasted with the fact that for the error-term we know for the prime counting function the gap betwwen unconditional known and under Riemann Hypothesis is much larger (there is not even an uncondition $O(x^{\theta})$ result for a $\theta$ less than $1$).
In general, gaps between primes are a bit of a different type of question, sure they also can be encoded via the prime-counting function, but by vague analogy they are rather questions on the "derivative" of the prime counting function, and knowledge of "extreme values" of a "derivative" (extremal large gaps, so how 'small' is this "derivative" sometimes) does not give that much information on the global behavior of the function (and RH is 'just' a statement on the approximate global behavior of the prime-counting function).
