Riemann Hypothesis on Grönwall's Function

Grönwall's function G is defined for all natural numbers n > 1 by G ( n ) = σ ( n ) n ⋅ loglog n where σ ( n ) is the sum of the divisors of n and log is the natural logarithm. We require the properties of extremely abundant numbers, that is to say left to right maxima of n ↦ G ( n ). We also use the colossally abundant and hyper abundant numbers. There are several statements equivalent to the famous Riemann hypothesis. 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 ′ ). In addition, we prove that the Riemann hypothesis is true when there exist infinitely many hyper abundant numbers n with any parameter u > 1. We claim that there could be infinitely many hyper abundant numbers with any parameter u > 1 and thus, the Riemann hypothesis would be true.


Introduction
As usual σ(n) is the sum-of-divisors function of n ∑ d|n d, where d | n means the integer d divides n.In 1997, Ramanujan's old notes were published where it was defined the generalized highly composite numbers, which include the superabundant and colossally abundant numbers [1] .A natural Qeios, CC-BY 4.0 • Article, May 24, 2023   Qeios ID: ZJNVF8.5 • https://doi.org/10.32388/ZJNVF8.5  1/6 number n is called superabundant precisely when, for all natural numbers m < n σ(m) m < σ(n) n .
A number n is said to be colossally abundant if, for some ϵ > 0, Every colossally abundant number is superabundant [2] .Let us call hyper abundant an integer n for which there exists where log is the natural logarithm.Every hyper abundant number is colossally abundant [[3], pp. 255] .In 1913, Grönwall n⋅log log n for all natural numbers n > 1 [4] .We have the Grönwall's Theorem: where γ ≈ 0.57721 is the Euler-Mascheroni constant [4] .
Proposition 3.There are infinitely many colossally abundant numbers N such that G(N) > e γ when the Riemann hypothesis is false [[5], Proposition pp. 204].
There are champion numbers (i.e.left to right maxima) of the function n ↦ G(n): for all natural numbers 10080 ≤ m < n.A positive integer n is extremely abundant if either n = 10080, or n > 10080 is a champion number of the function n ↦ G(n).In 1859, Bernhard Riemann proposed his hypothesis [6] .Several analogues of the Riemann hypothesis have already been proved [6] .
Proposition 4. The Riemann hypothesis is true if and only if there exist infinitely many extremely abundant numbers [[7], ( ) We use the following property for the extremely abundant numbers: Proposition 5. Let N < N ′ be two consecutive colossally abundant numbers and n > 10080 is some extremely abundant number, then N ′ is also extremely abundant when satisfying N < n < N ′[ [7], Lemma 21 pp. 12].This is our main theorem Theorem 1.The Riemann hypothesis is true if and only if there exist infinitely many consecutive colossally abundant The following is a key Corollary.
Corollary 1.The Riemann hypothesis is true when there exist infinitely many hyper abundant numbers N ′ with any parameter u > 1.
Putting all together yields a new criterion for the Riemann hypothesis.Now, we can conclude with the following result: Theorem 2. The Riemann hypothesis is true.
Proof.Note also that, for all u > 0 [ [3], pp. 254]: and so, we claim that there could be infinitely many hyper abundant numbers with any parameter u > 1 and thus, the Riemann hypothesis would be true.◻

Central Lemma
Lemma 1.For two real numbers y > x > e: y x > logy logx .
Proof.We have y = x + ε for ε > 0. We obtain that We need to show that using the well-known inequality log(1 + x) ≤ x for x > 0. For x > e, we have In conclusion, the inequality Proof.Suppose there are not infinitely many consecutive colossally abundant numbers N < N ′ such that G(N) < G(N ′ ).
This implies that the inequality G(N) ≥ G(N ′ ) always holds for a sufficiently large N when N < N ′ is a pair of consecutive colossally abundant numbers.That would mean the existence of a single colossally abundant number N ″ ≥ 10080 such that G(n) ≤ G(N ″ ) for all natural numbers n > N ″ according to Proposition 2. Certainly, the existence of such single colossally abundant number N ″ is because of the Grönwall's function G would become decreasing on colossally abundant numbers starting from some single value.We use the Proposition 5 to reveal that under these preconditions, then there are not infinitely many extremely abundant numbers.This implies that the Riemann hypothesis is false as a consequence of Proposition 4. By contraposition, if the Riemann hypothesis is true, then there exist infinitely many consecutive Suppose that there exist infinitely many consecutive colossally abundant numbers N < N ′ such that G(N) < G(N ′ ).From these assumed infinitely many consecutive colossally abundant numbers N < N ′ such that G(N) < G(N ′ ), then there could be only a finite amount of these N ′ such that e γ < G(N ′ ) because of the Proposition 1 and the properties of limit superior.
Thus, we deduce there could be only a finite amount of colossally abundant numbers N ″ such that e γ < G(N ″ ).However, when the Riemann hypothesis is false, then there are infinitely many colossally abundant numbers N ″ such that e γ < G(N ″ ) by Proposition 3. Therefore, the Riemann hypothesis would be true when there exist infinitely many consecutive colossally abundant numbers The result is done.◻

Proof of Corollary 1
Proof.Suppose there exists a large enough hyper abundant numbers N ′ with a parameter u > 1.We know that N ′ must be also a colossally abundant number.Let N be the greatest colossally abundant number such that N < N ′ , which means that N and N ′ is a pair of consecutive colossally abundant numbers.By definition of hyper abundant, we have which is the same as Hence, it is enough to show that Finally, proof is complete by Theorem 1. ◻ Qeios, CC-BY 4.0 • Article, May 24, 2023 Qeios ID: ZJNVF8.5 • https://doi.org/10.32388/ZJNVF8.