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Riemann zeta function for nonnegative even integers. The Riemann zeta function for negative even integers is 0 (those are the trivial zeros of the Riemann zeta function). The Riemann zeta function for nonnegative even integers is given by (note that is the only rational number for nonnegative even integers) where are the Bernoulli numbers.
So a set having measure zero means that the set is somehow really small. It isn't too big. If it is too big then its measure will be a positive number. The relation to the Riemann integral is that if $f(x)$ doesn't have too many discontinuities on its domain of integration, then it is integrable.