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The Riemann Hypothesis is true if and only if Robin's Inequality is true for all n>5040. It has also been shown by Akbary and Friggstad that the smallest counterexample greater than 5040, if it exists, must be a Superabundant Number.

These two facts suggest that Robin's Inequality might provide a more elementary way to show the Riemann Hypothesis is true or help provide a counterexample in the form of a Superabundant Number which violates the inequality. Robin's Inequality is elegantly stated:

$$f(n)=\frac{\sigma (n)}{e^{\gamma} n \log \log n}<1$$

Below is a plot of the LHS for the first 2000 Superabundant Numbers above 5040 from A004394.

enter image description here

Looking at this plot and knowing how close the values get to 1 make it seem plausible that RH could be false, but if true it appears to be an asymptotically an increasing function bounded above by 1. Examining the remaining terms provided, the function behavior looks similar and for the largest term listed $f(a[1000000])\approx0.9998655$. No terms before the millionth cross the threshold of one and $a[1000000]\approx10^{103082}$.

Is this the highest known verification of RH? It is my understanding that ZetaGrid and others working with the zeta function directly have only verified zeros up to $10^{13}$. Also, from my understanding of Briggs he appears to have only verified completely up to $10^{154}$ using SA numbers?

In fact, based on this curve and the possible relationship shown here, I am willing to venture a conjecture.

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    $\begingroup$ appears you want Briggs projecteuclid.org/download/pdf_1/euclid.em/1175789744 $\endgroup$ – Will Jagy Jul 15 at 20:17
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    $\begingroup$ the bigger check was on CA numbers; note how he is not saying "the first" counterexample is CA. And he checked those very, very high. I show computation of CA numbers at math.stackexchange.com/questions/3288519/… $\endgroup$ – Will Jagy Jul 15 at 20:51
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    $\begingroup$ That the zeros of $\zeta(s)$ of imaginary part $< N=10^{18}$ are on the critical line implies $\psi(x)= x + O(x^{1/2} \log^4 x)$ effective bound is true for $x < f(N)$ and that the Robin inequality is true for $\log n < f(N)$ with $f(N)$ being something like $N^a$. @WillJagy $\endgroup$ – reuns Jul 15 at 23:41
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    $\begingroup$ A "more elementary" way to show the Riemann Hypothesis is true?!? Seriously, there is no reason at all to think RH should be accessible to a proof by Robin's inequality. If anything, the relation between them means Robin's inequality is going to be very hard to prove. Accumulating numerical evidence is not likely to lead to any actual proof. $\endgroup$ – user1728 Jul 19 at 18:48
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    $\begingroup$ Numerical evidence will never lead to a proof obviously, but observation is the first step in the scientific method and is often overlooked in mathematics. By "more elementary" I just mean it doesn't on the surface involve analytic number theory. You must be aware that some conjectures that were originally proven with complex machinery later turned out to have simpler elementary proofs. In short, I don't find your comment very constructive @user1728, perhaps this is why you choose to remain anonymous? As the immortal poet TS once said, "haters shall continueth to hate". $\endgroup$ – Goldbug Jul 19 at 19:57

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