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 $\sigma>1$.

Let $\Gamma$ be the gamma function and finally

Let $\ds \map \omega x = \sum_{n \mathop = 1}^\infty e^{- \pi n^2 x}$

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

Proof
The Gamma function $\Gamma: \C \to \C$ is defined, for the open right half-plane, as:

Therefore, if we set $z = \dfrac s 2$, we have:

Substituting $t = \pi n^2 x$ and $\rd t = \pi n^2 \rd x$ we have:

Therefore:

Summing over $n$ on both sides of the equation and assuming $s \in \C$ is a complex number with real part $\sigma>1$, we get:

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

Focusing our attention to the first integral on the right hand side of the equation above and making a U substitution

Let $u = \dfrac 1 x$ and $\rd u = -\dfrac 1 {x^2} \rd x$:

When $x = 0$ then $u = \infty$ and When $x = 1$ then $u = 1$

Recall the Jacobi theta function:

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

where $\mathcal F$ denotes the Fourier transform.

Therefore, by the Poisson Summation Formula:

Therefore:

We can now enter Line 3 into Line 2 to complete the proof:

Combining lines $1$ and $4$ above, we have:

as required.

Also see

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