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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.
Let be an arbitrary point in the th subinterval. is called a Riemann sum for a given function and partition, and the value is called the mesh size of the partition. If the limit of the Riemann sums exists as , this limit is known as the Riemann integral of over the interval . The shaded areas in the above plots show the lower and upper sums for a constant mesh size.