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:
 * $\ds \pi^{-s / 2} \map \Gamma {\frac s 2} \map \zeta s = -\frac 1 {s \paren {1 - s} } + \int_1^\infty \paren {x^{s / 2 - 1} + x^{-\paren {s + 1} / 2} } \map \omega x \rd x$

where:
 * $\Gamma$ is the gamma function
 * $\ds \map \omega x = \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}$

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 $\rd t = \pi n^2 \rd x$:

Let $u = \dfrac 1 x$;

Then:
 * $\rd u = -\dfrac 1 {x^2} \rd x$


 * $x = 0 \implies u = \infty$


 * $x = 1 \implies u = 1$

Hence:

Recall the Jacobi theta function:

Since $e^{-x^2}$ is a fixed point of the Fourier transform, we have:


 * $\map {\FF \sqbrk {e^{-\pi t^2 x} } } u = x^{-1 / 2} e^{-\pi u^2 / x}$

where $\FF$ denotes the Fourier transform

Therefore, by the Poisson Summation Formula:

Hence the result.

Also see

 * Analytic Continuation of Riemann Zeta Function using Jacobi Theta Function
 * Functional Equation for Riemann Zeta Function