Riemann Hypothesis on Grönwall's FunctionEasyChair Preprint 9117, version historyVersion  Date  Pages  Version notes 

1  October 24, 2022  5   2  January 3, 2023  6   3  April 22, 2023  6  Put explicitly the proof of the Riemann hypothesis.  4  April 30, 2023  6  We improved the Central Lemma  5  May 9, 2023  6  Set explicitly that the Riemann hypothesis is true in a theorem.  6  May 15, 2023  6  We changed the symbol ⪆ by > and simplified the abstract.  7  May 24, 2023  6  We changed the abstract and proof of Theorem 1  8  June 11, 2023  6  We improved the Proof of Theorem 1  9  June 15, 2023  7  We added the section Conclusions in order to finalize the manuscript and avoid future versions.  10  June 23, 2023  6  We provide more arguments to proof of Theorem 2.  11  June 28, 2023  6  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 ′)."  12  June 30, 2023  7  I'm trying to make clearer and clearer again the arguments.  13  June 30, 2023  7   14  July 3, 2023  6  We removed the hyper abundant number definition (we changed the abstract, keywords and content).  15  July 8, 2023  7  We improved proofs of theorem 1 and 2.  16  July 23, 2023  7  We provided stronger arguments in the last theorem. 
Keyphrases: Arithmetic Functions, Colossally abundant numbers, Extremely abundant numbers, Hyper abundant numbers, Riemann hypothesis 
