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

$\blacksquare$

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