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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 $∀n\ge p_k∃p∈\mathbb{P} | n \le p \lt (1 + ε)n$.

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.

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 $∀n\ge p_k∃p∈\mathbb{P} | n \le p \lt (1 + ε)n$.

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.

EDIT

By GHfromMO's answer, it's clear that $ε$ has a lower bound. But is this result anyway notable? If so, where and how can I publish it?

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.

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 $∀n\ge p_k∃p∈\mathbb{P} | n \le p \lt (1 + ε)n$.

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.

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

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?

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.