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[parent] Riemann $\Xi$ function (Definition)

The Riemann Xi function

$\displaystyle \Xi(s) = \pi^{-\frac{1}{2}s} \Gamma(\frac{1}{2}s) \zeta(s),$
(where $ \Gamma(s)$ is Euler's Gamma function and $ \zeta(s)$ is the Riemann zeta function), is the key to the functional equation for the Riemann zeta function.

Riemann himself used the notation of a lower case xi ($ \xi$). The famous Riemann hypothesis is equivalent to the assertion that all the zeros of $ \xi$ are real, in fact Riemann himself presented his original hypothesis in terms of that function.

Riemann's lower case xi is defined as

$\displaystyle \xi(s) = \frac{1}{2} s(s-1) \Xi(s).$



"Riemann $\Xi$ function" is owned by PrimeFan. [ full author list (3) | owner history (4) ]
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functional equation for the Riemann Xi function (Theorem) by rspuzio
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Cross-references: function, terms, hypothesis, real, equivalent, Riemann hypothesis, functional equation, Riemann zeta function, Euler's gamma function
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This is version 8 of Riemann $\Xi$ function, born on 2003-01-29, modified 2008-08-06.
Object id is 3943, canonical name is RiemannXiFunction.
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Classification:
AMS MSC: 11M06 (Number theory :: Zeta and $L$-functions: analytic theory :: $\zeta $)
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