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In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted ζ and is named after the mathematician Bernhard Riemann. When the argument s is a real number greater than one, the zeta function satisfies the equation
The same equation in s above also holds when s is a complex number whose real part is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the complex plane by analytic continuation, except for a simple pole at s = 1. The complex derivative exists in this more general region, making the zeta function a meromorphic function. The above equation no longer applies for these extended values of s, for which the corresponding summation would diverge. For example, the full zeta function exists at s = −1 , but the corresponding series would be 1 + 2 + 3 + … , {\textstyle 1+2+3+\ldots \,,} whose partial sums would grow indefinitely large.
The zeta function values listed below include function values at the negative even numbers , for which ζ = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the Riemann hypothesis.