Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function

Theorem
Let $\zeta$ be the Riemann zeta function.

Let $s \in \C$ be a complex number with real part $\sigma>1$.

Then:
 * $\displaystyle \pi^{-s / 2} \Gamma \left({\frac s 2}\right) \zeta \left({s}\right) = - \frac 1 {s \left({1 - s}\right)} + \int_1^\infty \left({x^{s / 2 - 1} + x^{- \left({s + 1}\right) / 2} }\right) \omega \left({x}\right) \ \mathrm d x$

where:
 * $\Gamma$ is the gamma function
 * $\displaystyle \omega \left({x}\right) = \sum_{n \mathop = 1}^\infty e^{- \pi n^2 x}$

Thus:
 * $\xi \left({s}\right) = \xi \left({1 - s}\right)$

where $\xi$ is the completed Riemann zeta function.

Proof
By definition of the Gamma function:


 * $\displaystyle \Gamma \left({\frac s 2}\right) = \int_0^\infty t^{s / 2 - 1} e^{-t} \ \mathrm d t$

First substitute $t = \pi n^2 x$ to give:


 * $\displaystyle \pi^{-s / 2} \Gamma \left({\frac s 2}\right) n^{-s} = \int_0^\infty x^{s / 2 - 1} e^{-\pi n^2 x} \ \mathrm d x$

By definition, on $\operatorname{Re} \left({s}\right) > 1$:


 * $\displaystyle \zeta \left({s}\right) = \sum_{n \mathop = 1}^\infty n^{-s}$

Therefore summing over $n \ge 1$:


 * $\displaystyle \pi^{-s / 2} \Gamma \left({\frac s 2}\right) \zeta \left({s}\right) = \int_0^\infty x^{s / 2 - 1} \omega \left({x}\right) \ \mathrm d x$

where:
 * $\displaystyle \omega \left({x}\right) = \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}$

Next the integral is split at $x = 1$ as follows:

Recall the Jacobi theta function:

Since $e^{-x^2}$ is a fixed point of the Fourier transform, by Properties of the Fourier Transform:


 * $\mathcal F \left[{e^{-\pi t^2 x} }\right] \left({u}\right) = x^{-1 / 2} e^{-\pi u^2 / x}$

where $\mathcal F$ denotes the Fourier transform.

Therefore, by the Poisson Summation Formula:


 * $\theta \left({x}\right) = \dfrac 1 {\sqrt x} \theta \left({\dfrac 1 x}\right)$

whence:


 * $\omega \left({\dfrac 1 x}\right) = -\dfrac 1 2 + \dfrac 1 2 \sqrt x + \sqrt x \omega \left({x}\right)$

Therefore:

So:


 * $\displaystyle \pi^{-s / 2} \Gamma \left({\frac s 2}\right) \zeta \left({s}\right) = -\frac 1 {s \left({1 - s}\right)} + \int_1^\infty \left[{x^{s / 2} + x^{\left(1 - s\right) / 2} }\right] \omega \left({x}\right) \frac {\mathrm d x} x$

as required.

The functional equation:


 * $\xi \left({s}\right) = \xi \left({1 - s}\right)$

follows upon observing that this integral is invariant under $s \mapsto 1 - s$.