I can prove that given $ε$ chosen arbitrarily small, if $\prod_{p \le p_k} p^{\frac{1}{p-1}} \lt \frac{1 + ε}{e} p_k$ then $∀n\gt p_k∃p∈\mathbb{P} | n \le p \lt (1 + ε)n$.

Actually this result is better than Bertrand's Postulate. And I've seen this paper which has a worser result.

But how much is this result notable? If so, how and where do I publish it?

EDIT

In my first post I've made a huge mistake: I've written a sum instead of a product! Now it's correct.