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functional equation for the Riemann Xi function
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(Theorem)
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The Riemann Xi Function satisfies the following functional equation:
This equation directly implies the Riemann Zeta function's functional equation.
This equation plays an important role in the theory of the Riemann Zeta function. It allows one to analytically continue the Zeta and the Xi functions to the whole complex plane. The definition of the Zeta function as a series is only valid when
 . By using this equation, one can express the values of these two functions when
 in terms of the values when
 . As an illustration of its importance, one can cite the fact that there are no zeros of the Zeta function with real part greater than 1, so without this functional equation the study of the Zeta function would be very limited.
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"functional equation for the Riemann Xi function" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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Cross-references: real part, terms, series, complex plane, Riemann zeta function, theory, function's, implies, equation, functional equation, Riemann Xi function
There are 2 references to this entry.
This is version 4 of functional equation for the Riemann Xi function, born on 2003-01-30, modified 2005-01-07.
Object id is 3946, canonical name is FunctionalEquationForTheRiemannXiFunction.
Accessed 2174 times total.
Classification:
| AMS MSC: | 11M06 (Number theory :: Zeta and $L$-functions: analytic theory :: $\zeta $) |
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Pending Errata and Addenda
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