Riemann zeta function

Summary

The Riemann zeta function is given by the series $\zeta(s) = \sum_{n \ge 1} n^{-s}$ for complex $s$ with $\operatorname{Re} s > 1$. It extends to a meromorphic function on $\C$ with a single pole at $s = 1$. Through the Euler product formula it encodes the prime numbers, and the location of its zeros controls the error term in the prime number theorem. The Riemann hypothesis about these zeros is one of the main open problems of mathematics.