Editing Riemann zeta function (section)

==Riemann's Xi function==
{{main|Riemann Xi function}}

Riemann also found a [[Symmetry|symmetric]] version of the functional equation by setting
<math display="block">\xi(s) =\frac{s(s-1)}{2} \times \pi^{-\frac{s}{2}}\Gamma\left( \frac{s}{2} \right)\zeta(s) =  (s-1)\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}+1\right)\zeta(s)\ ,</math>
which satisfies:
<math display="block"> \xi(s) = \xi(1 - s) ~.</math>

Returning to the functional equation's derivation in the previous section, we have

<math display="block">
\xi(s) =\frac12 + \frac{s(s-1)}{2} \int_1^\infty \left(x^{-\frac{s}{2}-\frac{1}{2}} + x^{\frac{s}{2}-1}\right)\psi(x) dx
</math>

Using [[integration by parts]], 
<math display="block">
\xi(s) =\frac12 - \left[\left(sx^{\frac{1-s}{2}} + (1-s)x^{\frac{s}{2}}\right)\psi(x)\right]_1^\infty + \int_1^\infty \left(sx^{\frac{1-s}{2}} + (1-s)x^{\frac{s}{2}}\right)\psi'(x) dx
</math>
<math display="block">
\xi(s) =\frac12 + \psi(1) + \int_1^\infty \left(sx^{\frac{1-s}{2}} + (1-s)x^{\frac{s}{2}}\right)\psi'(x) dx
</math>

Using integration by parts again with a factorization of <math>x^{\frac32}</math>,
<math display="block">
\xi(s) =\frac12 + \psi(1) - 2\left[x^{\frac32}\psi'(x)\left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right)\right]_1^\infty + 2\int_1^\infty \left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right)\frac{d}{dx}\left[x^{\frac32}\psi'(x)\right] dx
</math>
<math display="block">
\xi(s) =\frac12 +\psi(1) + 4\psi'(1) + 2\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]\left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right) dx
</math>

As <math>\frac12 +\psi(1) + 4\psi'(1)=0</math>,

<math display="block">
\xi(s) = 2\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]\left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right) dx
</math>

Remove a factor of <math>x^{-\frac14}</math> to make the exponents in the remainder opposites. 

<math display="block">
\xi(s) = 2\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]x^{-\frac14}\left(x^{\frac{s-\frac12}{2}} + x^{\frac{\frac12-s}{2}}\right) dx
</math>

Using the [[hyperbolic functions]], namely <math>\cos(x)=\cosh(ix)=\frac{e^{ix}+e^{-ix}}{2}</math>, and letting <math>s=\frac12+it</math> gives
<math display="block">
\xi(s) = 4\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]x^{-\frac14}\cos(\frac{t}2\log x) dx
</math>

and by separating the integral and using the [[power series]] for <math>\cos</math>,
<math display="block">
\xi(s) = \sum_{n=0}^\infty a_{2n}t^{2n}
</math>
which led Riemann to his famous hypothesis.