Is there some numerical evidence that $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ for any large enough $ x $?

Question

Disclaimer : the following reasoning is a physicist's one and as such may not be suitable for this website. Still it may give rise to potentially interesting insights so I ask it anyway.

I stumbled upon an article whose abstract claims that for any sufficiently large $ x $ and any $ \delta>0.525 $, one has $\pi(x+x^{\delta})-\pi(x)\gg_{k}\dfrac{x^{\delta}}{\log^{k}x}$
where $\pi(y)$ is the number of primes less than or equal to the real number $y$.

My idea is to make the rhs equal to 1, so that one would formally get :

$ x^{\delta}\approx\log^{k}x $

Hence $ \delta\approx\dfrac{k\log\log x}{\log x} $ .

As the latter quantity attains a maximum of $\delta=1/e$ (as can be easily checked by calculus), this leads, assuming $ k $ can be taken equal to 1, to the following conjecture :

For any large enough $ x $, one has $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ .

Note that this is stronger than what can be obtained assuming RH, but weaker than Cramer's conjecture.

My question is : is the considered conjecture supported by numerical computation ? Is there a better (hence "more rigorous") heuristics that leads to the same conclusion ?

Answer 1

Turning my comment into an answer: your conjecture is supported by numerical computation, but much stronger ones are also supported: for instance, Cramér's conjecture, based on models of the primes as a 'random set', suggests (with some slight modifications to curb its optimism) that the largest prime gaps are $O\left((\log n)^{2+\epsilon}\right)$: much smaller than $x^{1/e}$, and in fact smaller than $x^\epsilon$ for any $\epsilon$. Terrence Tao has some discussion of these models that you might be interested in.

On the flip side, even though your conjecture is probably much weaker than the actual asymptotics, it's beyond the current reach of provability; AFAIK the best known result without the RH is that there's a prime between $x$ and $x+x^{0.525}$, and improvements on that value come incredibly slowly; even getting it down to $x^{1/2}$ would be an incredible advance and (from what I understand of the way the bound has been improved over time) would take entirely new techniques.