Calculate the max gap size out of the Firoozbakht's conjecture

Question

You can find the following image at the Wikipedia page for the Firoozbakht's conjecture. The conjecture states that $p_n^{1/n}$ is a strictly decreasing function. How can one calculate the gap size out of the conjecture? Or how is the Firoozbakht's conjecture connected to the prime gaps?

Answer 1

I'll prove the statements in Wikipedia article, which is

if $p_n^{1/n}$ is decreasing function, then $g_n < (\log p_n)^2-\log p_n$ for all $n>4$ and $g_n < (\log p_n)^2-\log p_n-1$ for all $n>9$

Lemma. if $x>10$, $x^{(1+1/x)}-x$ grows slower than $(\log x)^2-\log x+C$ for any constant $C$.

Proof of lemma.

$\frac{d}{dx}(x^{(1 + 1/x)} - x) = x^{(1/x + 1)} (x+1-\log x)/x^2 - 1$ and $\frac{d}{dx}\left((\log x)^2-\log x+C\right)=(2\log x-1)/{x}$.

So it is proving $$\frac{x^{(1/x + 1)} (x+1-\log x)}{x^2} - 1\lt\frac{2\log x-1}{x}$$

Some processing yields $${x^{(1/x)} (x+1-\log x)}\lt x+2\log x-1$$

taking log on each side $$\frac{\log x}{x}<\log \left( 1+\frac{3\log x-2}{x+1-\log x}\right)$$

which is true because if $x>10$, $\log x/x < 1/2$ and $$\log \left( 1+\frac{3\log x-2}{x+1-\log x}\right)> \log \left( 1+\frac{2\log x}{x}\right)>\frac{2\log x}{x}-\frac{1}{2}\frac{4\log ^2x}{x^2}>\frac{\log x}x$$

Done!

Now note that if $p_n^{1/n}$ is decreasing, then $p_{n+1} < p_n^{1+1/n}$ and $g_n < p_n^{1+1/n}-p_n$. By lemma, it suffices to check that the inequalities hold for $n=5$ and $n=10$ respectively.

Answer 2

Firoozbakht's conjecture is connected to the size of prime gaps in two ways.

(1) If prime gaps are "not too large", then Firoozbakht's conjecture is true: $$ p_{n+1}-p_n < \log^2 p_n - \log p_n - 1.17, \ \ n>9 \quad\Rightarrow\quad p_{n+1}<p_n^{1+1/n}. $$ This is Theorem 3 in arXiv:1506.03042 (J. Integer Sequences, 18, 2015, Article 15.11.2).

(2) If Firoozbakht's conjecture is true, then prime gaps are "not too large": $$ p_{n+1}<p_n^{1+1/n}, \ \ n>9 \quad\Rightarrow\quad p_{n+1}-p_n < \log^2 p_n - \log p_n - 1. $$ This is Theorem 1 in arXiv:1506.03042.