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# functional equation for the Riemann Xi function

The Riemann Xi Function satisfies the following functional equation:

$\Xi(s)=\Xi(1-s)$ |

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 $\Re(s)>1$. By using this equation, one can express the values of these two functions when $\Re(s)<1$ in terms of the values when $\Re(s)>1$. 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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11M06*no label found*

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## Comments

## tiny nitpick

Isn't the lower-case xi the traditional notation? In tex, it would be \xi rather than \Xi.

Aside, Riemann himself stated his famous conjecture (Riemann hypothesis) in terms of xi, not in terms of zeta. In those terms, the conjecture is that all the zeros of xi are real. It is also suspected that all those zeros are simple.

No big deal anyway.

## Re: tiny nitpick

Yes, I did know that, it really depends who you read. I do have a copy of Riemann's original paper, and he does use the lower case xi, but in most of my textbooks they use the uppercase one, so there is an ambiguity (hence I decided to use the upper case version because I think it looks better ;))