Riemann zeta function

In mathematics, the Riemann zeta function, named after Bernhard Riemann, is a meromorphic function defined for complex numbers with imaginary part $\scriptstyle \Im (s)>1$ 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 whole set of positive 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.