Riemann zeta function

The Riemann zeta function ζ(s) is defined for any complex number s with real part > 1 as:

             ∞   1
    ζ(s)  =  ∑  ----
             n=1  ns

In the region { s : Re(s)>1 }, this infinite series converges and defines a holomorphic function (see complex analysis).

The connection between this expression and prime numbers was already realized by Leonhard Euler:

                  1
    ζ(s)  =  ∏  ------
             p  1 - p-s

an infinite product extending over all prime numbers p. This is a consequence of the formula for the geometric series and the fundamental theorem of arithmetic.

[Bernhard Riemann]? realized that the zeta function can be extended in a unique way to a holomorphic function ζ(s) defined for all complex numbers s with s ≠ 1. It is this function that is the object of the Riemann hypothesis.

The zeros of ζ(s) are important because certain path integrals of the function ln(1/ζ(s)) can be used to approximate the prime counting function π(x) (see prime number theorem).