That is, each member of the list is a product of distinct primes. Call an integer red if it is the product of an even number of distinct primes, and blue if it is the product of an odd number of distinct primes. Of the squarefree integers up to 30, exactly 8 are red and 11 are blue. There are only 3 more blue numbers than red numbers. The Riemann Hypothesis says roughly that, of the squarefree integers up to an arbitrarily large number N, the discrepancy between the quantities of red numbers and blue numbers is never very large.
Computers can only work with finite numbers in a finite amount of time.
The Riemann Hypothesis concerns arbitrarily large and arbitrarily small numbers, which necessitates an understanding of infinitely many cases. Hence, while computers can help to provide insight, they do not provide the rigorous, rock-solid proofs that mathematicians seek in their work. There are other equivalent, less silly, ways of expressing the Riemann Hypothesis. Suffice to say that, if someone devised a proof, then much more would be known about the distribution of primes numbers than currently. Note that Fermat's Last Theorem is not on the Clay Institute list.