Riemann zeta function

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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). 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 connection between this function 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.

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). At certain points the zeta function takes on values which can be written in closed form, e.g. ζ(2) = π2/6 and ζ(4) = π4/90, which give well-known infinite series for π.