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 is maximal for $ x=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 ?

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 ?

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} $.

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 is maximal for $ x=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 ?

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 is maximal for $ x=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 ?