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Riemann Zeta Function 📂Functions

Riemann Zeta Function

Definition

The function defined as ξ\xi is called the Riemann xi Function. ξ(s):=12s(s−1)π−s/2ζ(s)Γ(s2) \xi (s) := {{ 1 } \over { 2 }} s ( s-1) \pi^{-s/2} \zeta (s) \Gamma \left( {{ s } \over { 2 }} \right)

Explanation

The Riemann xi function was originally defined in a slightly different form, but Edmund Landau redefined it with the lowercase xi ξ\xi and the original Riemann xi function is defined as Ξ(z):=ξ(12+zi)\Xi (z) := \xi \left( {{ 1 } \over { 2 }} + zi \right) using the uppercase Ξ\Xi1.

Theorem

ξ(1−s)=ξ(s) \xi ( 1 - s) = \xi (s) Meanwhile, the Riemann xi function is symmetric about s=12\displaystyle s = {{ 1 } \over { 2 }}, which, according to the original definition of the Riemann xi function, could also better represent the symmetry as follows. Ξ(−z)=Ξ(z) \Xi ( -z ) = \Xi ( z )

Proof2

Part 1.

Γ(x)=∫0∞tx−1e−tdt \Gamma (x) = \int_{0}^{\infty} t^{x-1} e^{-t} dt From the definition of the Gamma Function, let t=n2πzt = n^{2} \pi z Γ(x)=∫0∞(n2πz)x−1e−n2πzn2πdz=n2π(n2π)x−1∫0∞zx−1e−n2πzdz \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 x:=s2\displaystyle x := {{ s } \over { 2 }} n−sπ−s/2Γ(s2)=∫0∞zs2−1e−n2πzdz 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 ∑n∈N\sum_{n \in \mathbb{N}}, from the definition of the Riemann Zeta Function, when Re⁡(s)>1\re(s) > 1 ζ(s)π−s/2Γ(s2)=∑n∈Nn−sπ−s/2Γ(s2)=∑n∈N∫0∞zs2−1e−n2πzdz=∫0∞zs2−1∑n∈Ne−n2πzdz \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: ϑ(τ):=∑n∈Ze−πn2τ \vartheta (\tau) := \sum_{n \in \mathbb{Z}} e^{-\pi n^{2} \tau } The function defined as ϑ\vartheta is called the Jacobi Theta Function, and it has the following properties. ϑ(τ)=1τϑ(1τ) \vartheta ( \tau ) = \sqrt{ {{ 1 } \over { \tau }}} \vartheta \left( {{ 1 } \over { \tau }} \right)

By dividing the integration interval into [0,1)[0,1) and [1,∞)[1 , \infty) and substituting as in τ:=1z\tau := {{ 1 } \over { z }} for [0,1)[0,1), since dz=∣1τ2∣dτdz = \left| {{ 1 } \over { \tau^{2} }} \right| d \tau π−s/2ζ(s)Γ(s2)=∫0∞zs2−1ϑ(z)dz=∫01zs2−1ϑ(z)dz+∫1∞zs2−1ϑ(z)=∫1∞τ1−s2ϑ(1τ)1τ2dτ+∫1∞zs2−1ϑ(z)dz=∫1∞τ−1−s2ϑ(1τ)dτ+∫1∞zs2−1ϑ(z)dz=∫1∞τ−1−s2τϑ(τ)dτ+∫1∞zs2−1ϑ(z)dz=∫1∞τ−s2−12ϑ(τ)dτ+∫1∞zs2−1ϑ(z)dz \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 dzdz again π−s/2ζ(s)Γ(s2)=∫1∞[z−s2−12+zs2−1]ϑ(z)dz \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−s1-s instead of ss π−(1−s)/2ζ(1−s)Γ(1−s2)=∫1∞[z−1−s2−12+z1−s2−1]ϑ(z)dz=∫1∞[zs2−1+z−s2−12]ϑ(z)dz=π−s/2ζ(s)Γ(s2) \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 (1−s)((1−s)−1)2=s(s−1)2\displaystyle {{ (1-s) ((1-s)-1) } \over { 2 }} = {{ s (s-1) } \over { 2 }} (1−s)((1−s)−1)2π−(1−s)/2ζ(1−s)Γ(1−s2)=s(s−1)2π−s/2ζ(s)Γ(s2) {{ (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 ξ(1−s)=ξ(s) \xi ( 1 - s) = \xi (s)

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  1. https://en.wikipedia.org/wiki/Riemann_Xi_function ↩︎

  2. https://en.wikipedia.org/wiki/Riemann_zeta_function ↩︎