Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function/Lemma 1
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Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function: Lemma 1
- $\ds \pi^{-s / 2} \map \Gamma {\frac s 2} \map \zeta s = \int_0^1 x^{s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x } \rd x + \int_1^\infty x^{s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x } \rd x$
where:
- $\map \Gamma s$ is the gamma function
- $\map \zeta s$ is the Riemann zeta function
- $s \in \C$ is a complex number with real part $s > 1$
- $x \in \R_{>0}$.
Proof
The gamma function $\Gamma: \C \to \C$ is defined, for the open right half-plane, as:
- $\ds \map \Gamma z = \int_0^{\infty} t^{z - 1} e^{-t} \rd t$
Setting $z = \dfrac s 2$:
- $\ds \map \Gamma {\dfrac s 2} = \int_0^{\infty} t^{s/2 - 1} e^{-t} \rd t$
Substituting $t = \pi n^2 x$ and $\rd t = \pi n^2 \rd x$:
\(\ds \map \Gamma {\dfrac s 2}\) | \(=\) | \(\ds \int_0^\infty \paren {\pi n^2 x}^{s/2 - 1} e^{-\paren {\pi n^2 x} } \pi n^2 \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^\infty \pi ^{s/2} n^s x^{s/2 - 1} e^{-\pi n^2 x} \rd x\) | Power of Product, Power of Power and Product of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds \pi^{s/2} n^s \int_0^{\infty} x^{s/2 - 1} e^{-\pi n^2 x} \rd x\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \pi^{-s/2} \map \Gamma {\dfrac s 2} n^{-s}\) | \(=\) | \(\ds \int_0^\infty x^{s/2 - 1} e^{-\pi n^2 x} \rd x\) | multiplying both sides by $\pi^{-s/2} n^{-s}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \pi^{-s/2} \map \Gamma {\dfrac s 2} \sum_{n \mathop = 1}^\infty n^{-s}\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \int_0^\infty x^{s/2 - 1} e^{-\pi n^2 x} \rd x\) | summing over $n$ and assuming $s \in \C: \map \Re s > 1$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^\infty x^{s/2 - 1} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x} \rd x\) | Fubini's Theorem | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \pi^{-s/2} \map \Gamma {\dfrac s 2} \map \zeta s\) | \(=\) | \(\ds \int_0^\infty x^{s/2 - 1} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x} \rd x\) | Definition of Riemann Zeta Function | ||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 x^{s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x} \rd x + \int_1^\infty x^{s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x} \rd x\) | Linear Combination of Definite Integrals |
$\blacksquare$