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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.
Two facts about these functions are: 1  if is a monotonic function defined on an interval, then is differentiable almost everywhere on, i.e. the set of numbers in such that is not differentiable in has Lebesgue measure zero. 2  if is a monotonic function defined on an interval, then is Riemann integrable.