
Glossary: Prime Pages: Top 5000: 
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 wellknown extension of the factorial function ((n+1) = n! for nonnegative 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)
Chris K. Caldwell © 19992017 (all rights reserved)
