The Riemann zeta function, introduced by Leonard Euler, is defined by
where the sum is over all positive integers n.
Euler also introduced a multivariate generalization of the zeta function
where the sum is over all decreasing k-tuples of positive integers. This generalized zeta function satisfies the following beautiful identity:
The multivariate zeta function and identities such as the one above are important in number theory and are the subject of open conjectures.