∞ 1 ζ(s) = ∑ ---- n=1 nsIn 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-san 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).