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

Theorem

Let $s \in \C: \map \Re s > 1$.

Let $x \in \R_{>0}$.

Then:

$\ds \pi^{-s / 2} \map \Gamma {\frac s 2} \map \zeta s = -\frac 1 {s \paren {1 - s} } + \dfrac 1 2 \int_1^\infty \paren {x^{s / 2 - 1} + x^{-\paren {s + 1} / 2} } \paren {\map {\vartheta_3} {0, e^{-\pi x} } - 1} \rd x$

where:

$\map \Gamma s$ is the gamma function

$\map \zeta s$ is the Riemann zeta function

$\ds \map {\vartheta_3} {0, e^{-\pi x} }$ is the Jacobi theta function of the third type.

Proof

Lemma 1

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

$\Box$

Lemma 2

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

$\Box$

Lemma 3

$\ds \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x} = \dfrac 1 2 \paren {\map {\vartheta_3} {0, e^{-\pi x} } - 1}$

$\Box$

\(\ds \pi^{-s / 2} \map \Gamma {\frac s 2} \map \zeta s\)

\(=\)

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

Lemma $1$

\(\ds \)

\(=\)

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

Lemma $2$

\(\ds \)

\(=\)

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

Linear Combination of Definite Integrals

\(\ds \)

\(=\)

\(\ds -\frac 1 {s \paren {1 - s} } + \frac 1 2 \int_1^\infty \paren {x^{s / 2 - 1} + x^{-s / 2 - 1 / 2} } \paren {\map {\vartheta_3} {0, e^{-\pi x} } - 1} \rd x\)

Lemma $3$

$\blacksquare$

Also see

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