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Can computers prove hypothesises or theorems?
In 1976, for example, Appel and Haken proved the famous Four-Color Problem (formulated in 1852) by using three computers to do the really hard parts. In fact, their proof could not even be checked without computers.
ZetaGrid verifies Riemann's Hypothesis for a large range by computers. But most of all, ZetaGrid summarized the distribution of the verified zeros in form of statistical data which contains interesting zero-patterns and at moment unproved heuristics.
Ergo: Various ways can give hints for a proof of mathematical problems.
How probable is a disproof of Riemann Hypothesis?
Most scientists (also Sebastian Wedeniwski, the author of ZetaGrid) believe that the Riemann Hypothesis is true. But in the history of the mathematics, there exist many long-standing hypothesises and conjectures which were proved to be wrong. For example, the truth of the Mertens Conjecture (formulated in 1897) would imply the Riemann Hypothesis. But 88 years later (1985), this conjecture was proved false by Odlyzko and te Riele. It is believed that there is no counterexample to Mertens Conjecture for arguments less than 1020.
An example for a close pair of zeros:
In previous computations, Jan van de Lune found two zeros with a distance less than 0.00007356 in the interval [1239587702.54745, 1239587702.54753):
 

 
Such close zeros are important, for example, for the de Bruijn-Newman constant (see a paper of Odlyzko).