User blog:Vel!/Proof of the Riemann hypothesis?

In this post I will sketch a possible proof of the. In 2002, it was proved that the Riemann hypothesis is equivalent to the inequality

\[\sigma(n) \leq H_n + \ln (H_n) e^{H_n}\]

where \(\sigma(n)\) is the (the number of divisors of \(n\)) and \(H_n\) is the  \(1 + 1/2 + 1/3 + \ldots + 1/n\). By showing that the inequality is true, it holds that the Riemann hypothesis is also true.

Lemma 1
Statement: \(\sigma(n) \leq n\).

Proof: If \(d\) is a divisor of \(n\), then \(1 \leq d \leq n\) and only \(n\) distinct values for \(d\) are possible.

Lemma 2
Statement: \(n \uparrow\uparrow 3 \leq n \uparrow\uparrow\uparrow 2\) for \(n \geq 3\).

Proof: By induction:

\[n \uparrow\uparrow 3 \leq n \uparrow\uparrow 2\]

\[e^{n \uparrow\uparrow 3} \leq e^{n \uparrow\uparrow 2}\]

\[\arctan(e^{n \uparrow\uparrow 3}) \leq \arctan(e^{n \uparrow\uparrow 2})\]

\[\arctan(e^{n \uparrow\uparrow 3})^{1/\arcsec \int_1^\infty \frac{dx}{\ln x}} \leq \arctan(e^{n \uparrow\uparrow 2})^{1/\arcsec \int_1^\infty \frac{dx}{\ln x}}\]