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