Riemann zeta function

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In mathematics, the Riemann zeta function, named after Bernhard Riemann, is a meromorphic function defined for real numbers s > 1 by the infinite series

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

(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.