Riemann Hypothesis on Grönwall's Function

Make clear the arguments

Put explicitly the proof of the Riemann hypothesis.

We improved the Central Lemma

Set explicitly that the Riemann hypothesis is true in a theorem.

We changed the symbol ⪆ by > and simplified the abstract.

We changed the abstract and proof of Theorem 1

We improved the Proof of Theorem 1

We added the section Conclusions in order to finalize the manuscript and avoid future versions.

We provide more arguments to proof of Theorem 2.

Today, a researcher has suggested me this change that I applied, he told me:

What does mean
" We state that the Riemann hypothesis is true if and only if there exist infinitely many consecutive colossally abundant numbers N < N ′ such that G(N ) < G(N ′)."
Do you mean
"We state that the Riemann hypothesis is true if and only if there exist infinitely many pairs (N,N') of consecutive colossally abundant numbers N < N ′ such that G(N ) < G(N ′)."

I'm trying to make clearer and clearer again the arguments.

We improved Theorem 2

We removed the hyper abundant number definition (we changed the abstract, keywords and content).

We improved proofs of theorem 1 and 2.

We provided stronger arguments in the last theorem.