an observation of Chudakov is mentioned :

For every $\theta>\frac{3}{4}$ there exists an $N$ such that $g_n<p_n^{\theta}$ for all $n\ge N$.

Questions :

Is there en efficient method to determine $N$ given $\theta$ ?

Is there any better result (with a concrete upper bound ; no constant or $O$-notation involved) , which has been proven without needing unproven conjectures (such as the riemann hypothesis) ?

The tightest bound (but not proven) is mentioned here :


The claim is : $g_n<\ln^2(p_n)-ln(p_n)-1$ for $n>9$, but unfortunately it relies on an unproven conjecutre mentioned in the article.

Besides the fact that the conjecture has been proven for $p\le 4\cdot 10^{18}$, is there a heuristic supporting the conjecture ?


(A) Is there an efficient method to determine $N$ for given $\theta$?

In general, for an arbitrary $\theta$, no such method is known. However, for $\theta\ge 0.525$ there is a result by Baker, Harman, Pintz which includes $O$-notation. Computations strongly suggests that for $\theta=0.525$ we can take $p_N=127$, i.e. $N=31$; and prime gaps starting at primes $p_n\ge127$ are invariably below $p_n^{0.525}$.

For the special case you mentioned, $\theta=3/4$, there is an apparent consensus that $N=1$; see e.g. this seqfan discussion. That is, a prime gap starting at $p_n$ is below $p_n^{3/4}$. Again, no rigorous proof has been published for this, but the confidence level I think is very high, on par with the Goldbach conjecture.

(B) Heuristics behind Firoozbakht's conjecture:

In Cramer's probabilistic model of primes, the maximal prime gap $G(p)$ starting at $p$ satisfies the condition $$ G(p) < \log^2 p - \log p - (1+\varepsilon) $$ with probability $1$. That is, almost all maximal prime gaps are expected to satisfy this condition in Cramer's model. This is mentioned (in a slightly simpler form) in OEIS A111943. (This still leaves us with the possibility that an infinite subsequence of maximal gaps do not satisfy the above inequality.)


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