The Riemann zeta function is a complex function denoted as ζ(s), where s is a complex number. It is defined for values of s with a real part greater than 1 as the infinite series ζ(s) = 1^(-s) + 2^(-s) + 3^(-s) + ..., and it can be analytically continued to other values, except for s = 1, where it has a simple pole. This function plays a crucial role in number theory, particularly in understanding the distribution of prime numbers. The famous Riemann Hypothesis conjectures that all non-trivial zeros of the zeta function lie on the critical line where the real part of s is 1/2, which has significant implications for the field.