Zeta-function is the name given to certain functions of the complex variable s = + it that play a fundamental role in analytic number theory. The most important example is the Riemann zeta-function z(s). In the right half plane {s ∈ C: 1 < } the Riemann zeta-function is defined by the infinite series

It can be shown that the infinite series defining the zeta-function converges absolutely and uniformly on all compact subsets of {s ∈ C: 1 < } and therefore the Riemann zeta-function is an analytic function in this domain. Series of this type are called Dirchlet series. More generally, if a(1), a(2), a(3), ... is a sequence of complex numbers then

is the associated Dirichlet series. The natural domain of convergence for such a series is always a right half plane, but the half plane may be empty or all of C. If the half plane is not empty then the series defines...