If the Riemann hypothesis is false, then there has to be a first counterexample for $\zeta(z)=0$ in the critical strip with $\Re(z) \ne \frac{1}{2}$. For such a counterexample, how large would $T=|\Im(z)|$ have to be?
On the one hand, we've only computed the first 10 trillion or so zeros, so it could be the very next one. On the other hand, assuming Odlyzko comments below are correct, there is no hope for disproving the RH by actually computing a counterexample. Andrew Odlyzko is quoted in "Prime Obsession" (by John Derbyshire, highly recommended book),
"For the entire range for which zeta has been studied--which is to say, for arguments on the critical line up to around 10^23 -- S mainly hovers between -1 and 1. The largest known value is around 3.2. There are strong reasons to believe that if S were ever to get up around 100, then the RH might be in trouble. The operative word is "might"; S attaining a value near 100 is a necessary condition for the RH to be in trouble, but not a sufficient one."
Could values of the S function ever get that big? "...Atle Selberg proved in 1946 that S is unbounded... Probably around t equals $10^{10^{10000}}$"
So then the first counterexample cannot be "the next" zero. Is this rigorously provable? Or could the S function jump immediately to values near 100 for the very next zero which isn't on the critical line? I'm also looking for a link to Selberg's S function.
edit The S function is just the log of zeta in the critical strip. Since $\zeta(z)$ goes to 1 as $\Re(z)$ increases, we take the logarithm from the positive side. here's a link: http://www.math.sjsu.edu/~goldston/MyS(t).pdf
$$S(T) = \frac{1}{\pi }\arg\zeta(\frac{1}{2}+iT) $$
The wikipedia artcle talks about the growth and average magnitude of S(T) as well. $S(T)$ jumps by $\pm1$ at zeros of the $\zeta$ function; and mainly hovers between -1 and 1; from Derbyshire's book, the largest known value is 3.2 So why would the magnitude of S(T) be connected with possible counterexamples to the RH? I know at a counter example, $S(T)$ would jump by $\pm 2$. Anyway, this is all new stuff to me; Gram points, S(T); fun stuff.