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The Riemann Hypothesis is equivalent to the statement: $$|\pi(x)-{\rm li}(x)|\le \frac {1}{8\pi}\sqrt {x}\log (x)\text { for all }x \geq 2657,\text{ (Schoenfeld, 1976)} $$ Which can be visually represented in the plot: enter image description here $$\text{with }\frac{1}{8\pi}\sqrt{x}\log(x)-|\text{J}(x)-\text{li}(x)|,\text{ (blue) }\\ \frac{\sqrt{x}}{\log(x)},\text{ (red) }\\ |\pi(x)-\text{J}(x)|,\text{ (yellow) }\\ |\text{J}(x)-\text{li}(x)|,\text{ (green) }\\ \text{where J}(x)=\sum_{n=1}^\infty\frac{\mu(n)}{n}\text{li}(x^{1/n}); $$ and on a larger scale:

enter image description here

where it is not so difficult to imagine the oscillations crossing the green line eventually (Littlewood, 1914), but requires a leap of faith to say the least, that the oscillations will eventually cross the blue line. I believe I am correct in assuming that it is only the oscillations that are in question here, & not the general asymptotic of $J(x)$, but am willing to be corrected!

If the crossing of the oscillations of the green line begin at Skewe's number (whatever the actual value of that is), then if the RH is false, the magnitude of the number at which the oscillations cross the blue line must be significantly larger than that - which, even at the current rate of advances in computing, quantum computers being developed at some point in the future notwithstanding, I find it difficult to believe that the RH will be proven false in my lifetime (I am 37! - not factorial btw).

My question is, is the falsity of the RH (a) believable; and (b) ever provable (if, in the unlikely event that it is true) in terms of computing limits?

It seems to me, in the spirit of Hardy, to be an analogue of Russell's teapot analogy!

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    $\begingroup$ There is a hint in the use of the word 'hypothesis'. $\endgroup$
    – copper.hat
    Mar 2, 2014 at 22:18
  • $\begingroup$ @ copper.hat, funny! $\endgroup$
    – martin
    Mar 2, 2014 at 22:19
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    $\begingroup$ My impression from reading discussions of the Hypothesis is that the "computational effort" is only exploratory, in the sense of getting a better idea of the behavior of the various functions involved. (A "computational proof" is probably not realistic.) It is interesting that enough people believe the proposition is likely true that they have even taken the time to construct proofs of statements of the form "RH $ \ \Rightarrow \ X \ $" . Otherwise, though, we still have no idea whether we'll be able to tell David Hilbert, when we awaken him in 2443, whether the Hypothesis is true... $\endgroup$
    – colormegone
    Mar 3, 2014 at 0:28
  • $\begingroup$ @RecklessReckoner in fact there have been some results proved by assuming RH is true and then assuming it is false (with the result holding true in both cases). $\endgroup$
    – Cameron Williams
    Mar 5, 2014 at 20:47
  • $\begingroup$ "I am 37! Not factorial btw" made me lol $\endgroup$
    – qwr
    Mar 21, 2014 at 1:27

2 Answers 2

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Finding an $x$ that violates the given inequality is only one way to disprove the Riemann Hypothesis. Another way would simply be to find a (nontrivial) zero of the zeta function with real part not equal to $1/2$. So far we've "only" computed about ten trillion zeros. The first counterexample could be the very next one.

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Probably the Riemann Hypothesis is true, in which case its falsity would not be provable (and the believability of its falsity is more a matter of psychology than mathematics), whether by computation or otherwise.

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