Riemann zeta function
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000: Riemann extended the definition of Euler's zeta function zeta(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:
[Functional Eq]
where the gamma function gamma(s) is the well-known extension of the factorial function (gamma(n+1) = n! for non-negative integers n):
[def of gamma]
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

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