Riemann Zeta Function
Definition
The function defined as $\xi$ is called the Riemann xi Function. $$ \xi (s) := {{ 1 } \over { 2 }} s ( s-1) \pi^{-s/2} \zeta (s) \Gamma \left( {{ s } \over { 2 }} \right) $$
- $\zeta$ is the Riemann Zeta Function.
- $\Gamma$ is the Gamma Function.
Explanation
The Riemann xi function was originally defined in a slightly different form, but Edmund Landau redefined it with the lowercase xi $\xi$ and the original Riemann xi function is defined as $\Xi (z) := \xi \left( {{ 1 } \over { 2 }} + zi \right)$ using the uppercase $\Xi$1.
Theorem
$$ \xi ( 1 - s) = \xi (s) $$ Meanwhile, the Riemann xi function is symmetric about $\displaystyle s = {{ 1 } \over { 2 }}$, which, according to the original definition of the Riemann xi function, could also better represent the symmetry as follows. $$ \Xi ( -z ) = \Xi ( z ) $$
Proof2
Part 1.
$$ \Gamma (x) = \int_{0}^{\infty} t^{x-1} e^{-t} dt $$ From the definition of the Gamma Function, let $t = n^{2} \pi z$ $$ \begin{align*} \displaystyle \Gamma \left( x \right) =& \int_{0}^{\infty} \left( n^{2} \pi z \right)^{x-1} e^{-n^{2} \pi z} n^{2} \pi dz \\ =& n^{2} \pi \left( n^{2} \pi \right)^{x-1} \int_{0}^{\infty} z^{x-1} e^{-n^{2} \pi z} dz \end{align*} $$ Then let $\displaystyle x := {{ s } \over { 2 }}$ $$ n^{-s} \pi^{-s/2} \Gamma \left( {{ s } \over { 2 }} \right) = \int_{0}^{\infty} z^{{{ s } \over { 2 }}-1} e^{-n^{2} \pi z} dz $$ Taking both sides as $\sum_{n \in \mathbb{N}}$, from the definition of the Riemann Zeta Function, when $\Re(s) > 1$ $$ \begin{align*} \zeta (s) \pi^{-s/2} \Gamma \left( {{ s } \over { 2 }} \right) =& \sum_{n \in \mathbb{N}} n^{-s} \pi^{-s/2} \Gamma \left( {{ s } \over { 2 }} \right) \\ =& \sum_{n \in \mathbb{N}} \int_{0}^{\infty} z^{{{ s } \over { 2 }}-1} e^{-n^{2} \pi z} dz \\ =& \int_{0}^{\infty} z^{{{ s } \over { 2 }}-1} \sum_{n \in \mathbb{N}} e^{-n^{2} \pi z} dz \end{align*} $$
Part 2.
Definition and Properties of the Jacobi Theta Function: $$ \vartheta (\tau) := \sum_{n \in \mathbb{Z}} e^{-\pi n^{2} \tau } $$ The function defined as $\vartheta$ is called the Jacobi Theta Function, and it has the following properties. $$ \vartheta ( \tau ) = \sqrt{ {{ 1 } \over { \tau }}} \vartheta \left( {{ 1 } \over { \tau }} \right) $$
By dividing the integration interval into $[0,1)$ and $[1 , \infty)$ and substituting as in $\tau := {{ 1 } \over { z }}$ for $[0,1)$, since $dz = \left| {{ 1 } \over { \tau^{2} }} \right| d \tau$ $$ \begin{align*} \pi^{-s/2} \zeta (s) \Gamma \left( {{ s } \over { 2 }} \right) =& \int_{0}^{\infty} z^{{{ s } \over { 2 }}-1} \vartheta (z) dz \\ =& \int_{0}^{1} z^{{{ s } \over { 2 }}-1} \vartheta (z) dz + \int_{1}^{\infty} z^{{{ s } \over { 2 }}-1} \vartheta (z) \\ =& \int_{1}^{\infty} \tau^{ 1 - {{ s } \over { 2 }}} \vartheta \left( {{ 1 } \over { \tau }} \right) {{ 1 } \over { \tau^{2} }} d \tau + \int_{1}^{\infty} z^{{{ s } \over { 2 }}-1} \vartheta (z) dz \\ =& \int_{1}^{\infty} \tau^{ -1 - {{ s } \over { 2 }}} \vartheta \left( {{ 1 } \over { \tau }} \right) d \tau + \int_{1}^{\infty} z^{{{ s } \over { 2 }}-1} \vartheta (z) dz \\ =& \int_{1}^{\infty} \tau^{ -1 - {{ s } \over { 2 }}} \sqrt{\tau} \vartheta \left( \tau \right) d \tau + \int_{1}^{\infty} z^{{{ s } \over { 2 }}-1} \vartheta (z) dz \\ =& \int_{1}^{\infty} \tau^{ - {{ s } \over { 2 }} - {{ 1 } \over { 2 }}} \vartheta \left( \tau \right) d \tau + \int_{1}^{\infty} z^{{{ s } \over { 2 }}-1} \vartheta (z) dz \end{align*} $$ By expressing the integrand uniformly as $dz$ again $$ \pi^{-s/2} \zeta (s) \Gamma \left( {{ s } \over { 2 }} \right) = \int_{1}^{\infty} \left[ z^{ - {{ s } \over { 2 }} - {{ 1 } \over { 2 }}} + z^{{{ s } \over { 2 }}-1} \right] \vartheta \left( z \right) dz $$
Part 3. In the above equation, even if the variable is $1-s$ instead of $s$ $$ \begin{align*} \pi^{-(1-s)/2} \zeta (1-s) \Gamma \left( {{ 1-s } \over { 2 }} \right) =& \int_{1}^{\infty} \left[ z^{ - {{ 1-s } \over { 2 }} - {{ 1 } \over { 2 }}} + z^{{{ 1-s } \over { 2 }}-1} \right] \vartheta \left( z \right) dz \\ =& \int_{1}^{\infty} \left[ z^{{{ s } \over { 2 }}-1} + z^{ - {{ s } \over { 2 }} - {{ 1 } \over { 2 }}} \right] \vartheta \left( z \right) dz \\ =& \pi^{-s/2} \zeta (s) \Gamma \left( {{ s } \over { 2 }} \right) \end{align*} $$ Multiplying both sides by $\displaystyle {{ (1-s) ((1-s)-1) } \over { 2 }} = {{ s (s-1) } \over { 2 }}$ $$ {{ (1-s) ((1-s)-1) } \over { 2 }} \pi^{-(1-s)/2} \zeta (1-s) \Gamma \left( {{ 1-s } \over { 2 }} \right) = {{ s (s-1) } \over { 2 }} \pi^{-s/2} \zeta (s) \Gamma \left( {{ s } \over { 2 }} \right) $$ Expressing this as the Riemann xi function gives $$ \xi ( 1 - s) = \xi (s) $$
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