Riemann zeta function
The zeta-function is a function very important to the Riemann Hypothesis. The function is
The series is convergent iff
. Leonhard 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
.
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