Riemann zeta function

In mathematics, the Riemann zeta function, named after Bernhard Riemann, is one of the most important special functions. Its generalizations have important applications to number theory, arithmetic geometry, graph theory, and dynamical systems, to name a few examples. The Riemann zeta function in particular gained prominence when it was shown to have a connection with the distribution of the prime numbers. The most important result related to the Riemann zeta function is the Riemann hypothesis, which was the 8th of Hilbert's Problems, and is one of the seven Millenium Prize Problems presented by the Clay Institute of Mathematics. As such, anyone who determines its truth or falsity is entitled to $1 million (U.S.)

Definition

The Riemann zeta function is a meromorphic function defined for complex numbers with real part $\scriptstyle \Re (s)>1$ by the infinite series

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}$

and then extended to all other complex values of s except s = 1 by analytic continuation. The function is holomorophic everywhere except for a simple pole at s = 1.

Euler's product formula for the zeta function is

$\zeta (s)=\prod _{p\ \mathrm {prime} }{\frac {1}{1-p^{-s}}}$

(the index p running through the set of prime numbers).

The celebrated Riemann hypothesis is the conjecture that all non-real values of s for which ζ(s) = 0 have real part 1/2. The problem of proving the Riemann hypothesis is the most well-known unsolved problem in mathematics.