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# Riemann zeta function

Riemann extended the definition of Euler's zeta function (*s*) to all complex numbers

*s*(except the simple pole at

*s*=1 with residue one). Euler’s product definition of this function still holds if the real part of

*s*is greater than one. To help understand the values for other complex numbers, Riemann derived the functional equation of the Riemann zeta function:

where the gamma function (

*s*) is the well-known extension of the factorial function ((

*n*+1) =

*n*! for non-negative integers

*n*):

Here the integral holds if the real part of

*s*is greater than one, and the product holds for all complex numbers

*s*.

**See Also:** RiemannHypothesis, EulerZetaFunction

**Related pages** (outside of this work)

- The Riemann hypothesis with expanded information on this function

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