Riemann zeta function
The zeta-function is a function very important to the Riemann Hypothesis. The function is The series is convergent [[iff] . Euler showed that when , the sum is equal to . Euler also found that since every number is the product of a certain combination of prime numbers, the zeta-function can also be expressed as . By summing up each of these geometric series in parentheses, we have the following identity, the so called Euler Product: .
However, the most important properties of the zeta function are based on the fact that it extends to a meromorphic function on the full complex plane which is holomorphic except at , where there is a simple pole of residue 1. Let us see how this is done: First, we wish to extend to . To do this, we introduce the alternating zeta function , which is convergent on . (This follows from one of the standard convergence tests for alternating series.) We then have . We therefore have when .
The next step is the functional equation: Let . Then . This gives us a meromorphic continuation of to all of .