Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function/Lemma 2

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Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function: Lemma 2

$\ds \int_0^1 x^{s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x} \rd x = -\frac 1 {s \paren {1 - s} } + \int_1^\infty x^{-\paren {s + 1} / 2} \sum_{n \mathop = 1}^\infty e^{-\paren {\pi n^2 x} } \rd x$

where:

$s \in \C: \map \Re s > 1$
$x \in \R_{>0}$.


Proof

Let $u = \dfrac 1 x$.

Then:

\(\ds \d u\) \(=\) \(\ds -\dfrac 1 {x^2} \rd x\) Derivative of Reciprocal
\(\ds x \to 0^+\) \(\implies\) \(\ds u \to +\infty\)
\(\ds x = 1\) \(\implies\) \(\ds u = 1\)


Hence:

\(\ds \int_0^1 x^{s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x} \rd x\) \(=\) \(\ds \int_\infty^1 \paren {\dfrac 1 u }^{s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\dfrac {\pi n^2} u} \paren {-\paren {\dfrac 1 u}^2} \rd u\) Integration by Substitution: Definite Integral
\(\ds \) \(=\) \(\ds -\int_\infty^1 \paren {\dfrac 1 u }^{s / 2 + 1} \sum_{n \mathop = 1}^\infty e^{-\dfrac {\pi n^2} u} \rd u\) Product of Powers
\(\ds \) \(=\) \(\ds \int_1^\infty \paren {u^{-1 } }^{s / 2 + 1} \sum_{n \mathop = 1}^\infty e^{-\dfrac {\pi n^2} u} \rd u\) Reversal of Limits of Definite Integral
\(\ds \) \(=\) \(\ds \int_1^\infty u^{-s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\dfrac {\pi n^2} u} \rd u\) Power of Power
\(\ds \) \(=\) \(\ds \int_1^\infty x^{-s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\dfrac {\pi n^2} x} \rd x\) rewrite in terms of $x$


Recall the Fourier Transform of $e^{-t^2}$:

\(\ds \map {\hat f \sqbrk {e^{- t^2} } } u\) \(=\) \(\ds \sqrt \pi e^{-\paren {\pi u}^2}\) Fourier Transform of Gaussian Function
\(\ds \map {\hat f \sqbrk {e^{-\paren {t \sqrt {\pi x} }^2 } } } u\) \(=\) \(\ds \dfrac {\sqrt \pi} {\sqrt {\pi x} } e^{-\paren {\pi \frac u {\sqrt {\pi x} } }^2}\) Scaling Property of Fourier Transform, setting $t = t \sqrt {\pi x}$
\(\ds \map {\hat f \sqbrk {e^{-\pi t^2 x} } } u\) \(=\) \(\ds x^{-1 / 2} e^{-\pi u^2 / x}\)


Therefore, by the Poisson Summation Formula:

\(\ds \sum_{t \mathop \in \Z} \map f t\) \(=\) \(\ds \sum_{u \mathop \in \Z} \map {\hat f \sqbrk {e^{-\pi t^2 x} } } u\) Poisson Summation Formula
\(\ds \leadsto \ \ \) \(\ds \sum_{-\infty}^\infty e^{- \pi t^2 x}\) \(=\) \(\ds \sum_{-\infty}^\infty x^{-1 / 2} e^{-\pi u^2 / x}\)
\(\ds \leadsto \ \ \) \(\ds \sum_{n \mathop = -\infty}^{-1} e^{-\pi n^2 x} + 1 + \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}\) \(=\) \(\ds \dfrac 1 {\sqrt x} \paren {\sum_{u \mathop = -\infty}^{-1} e^{-\pi u^2 / x } + 1 + \sum_{u \mathop = 1}^\infty e^{-\pi u^2 / x} }\)
\(\ds \leadsto \ \ \) \(\ds 1 + 2 \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}\) \(=\) \(\ds \dfrac 1 {\sqrt x} \paren {1 + 2 \sum_{u \mathop = 1}^\infty e^{-\pi u^2 / x} }\)
\(\ds \leadsto \ \ \) \(\ds \sqrt x + 2 \sqrt x \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}\) \(=\) \(\ds 1 + 2 \sum_{u \mathop = 1}^\infty e^{-\pi u^2 / x}\) multiplying by $\sqrt x$
\(\ds \leadsto \ \ \) \(\ds -1 + \sqrt x + 2 \sqrt x \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}\) \(=\) \(\ds 2 \sum_{u \mathop = 1}^\infty e^{-\pi u^2 / x}\) rearranging
\(\ds \leadsto \ \ \) \(\ds \sum_{u \mathop = 1}^\infty e^{-\pi u^2 / x}\) \(=\) \(\ds -\dfrac 1 2 + \dfrac 1 2 \sqrt x + \sqrt x \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}\) dividing by $2$
\(\ds \leadsto \ \ \) \(\ds \int_1^\infty x^{-s / 2 - 1} \sum_{u \mathop = 1}^\infty e^{-\pi u^2 / x} \rd x\) \(=\) \(\ds \int_1^\infty x^{-s / 2 - 1} \paren {-\frac 1 2 + \frac 1 2 \sqrt x + \sqrt x \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x } } \rd x\)
\(\ds \) \(=\) \(\ds -\dfrac 1 2 \int_1^\infty x^{-s / 2 - 1} \rd x + \dfrac 1 2 \int_1^\infty x^{-s / 2 - 1 / 2} \rd x + \int_1^\infty x^{-s / 2 - 1 / 2} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x} \rd x\) Linear Combination of Definite Integrals and Product of Powers
\(\ds \) \(=\) \(\ds -\dfrac 1 2 \paren {\lim_{\gamma \mathop \to +\infty} \intlimits {-\frac 2 s x^{-s / 2} } 1 \gamma} + \dfrac 1 2 \paren {\lim_{\gamma \mathop \to +\infty} \intlimits {\frac 2 {1 - s} x^{\paren {1 - s} / 2} } 1 \gamma} + \int_1^\infty x^{-s / 2 - 1 / 2} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x} \rd x\) Primitive of Power
\(\ds \) \(=\) \(\ds -\dfrac 1 2 \paren {0 - \paren {-\frac 2 s} } + \frac 1 2 \paren {0 - \paren {\frac 2 {1 - s} } } + \int_1^\infty x^{-s / 2 - 1 / 2} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x} \rd x\)
\(\ds \) \(=\) \(\ds -\frac 1 s - \frac 1 {1 - s} + \int_1^\infty x^{-\paren {s + 1} / 2} \sum_{n \mathop = 1}^\infty e^{-\paren {\pi n^2 x} } \rd x\)
\(\ds \) \(=\) \(\ds \frac {-\paren {1 - s} - s} {s \paren {1 - s} } + \int_1^\infty x^{-\paren {s + 1} / 2} \sum_{n \mathop = 1}^\infty e^{-\paren {\pi n^2 x} } \rd x\) Addition of Fractions
\(\ds \leadsto \ \ \) \(\ds \int_0^1 x^{s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x} \rd x\) \(=\) \(\ds -\frac 1 {s \paren {1 - s} } + \int_1^\infty x^{-\paren {s + 1} / 2} \sum_{n \mathop = 1}^\infty e^{-\paren {\pi n^2 x} } \rd x\)

$\blacksquare$

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