Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function/Lemma 3

Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function: Lemma 3

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

where:

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

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

Proof

Recall the definition of the Jacobi theta function of the third type:

The Jacobi Theta function of the third type is defined for all complex $z$ by:

$\forall z \in \C: \ds \map {\vartheta_3} {z, q} = 1 + 2 \sum_{n \mathop = 1}^\infty q^{n^2} \cos 2 n z$

where:

$q = e^{i \pi \tau}$

$\tau \in \C$ such that $\map \Im \tau > 0$

\(\ds \map {\vartheta_3} {z, q}\)

\(=\)

\(\ds 1 + 2 \sum_{n \mathop = 1}^\infty q^{n^2} \cos 2 n z\)

Definition of Jacobi Theta Function of the Third Type

\(\ds \)

\(=\)

\(\ds 1 + 2 \sum_{n \mathop = 1}^\infty \paren {e^{i \pi \tau} }^{n^2} \map \cos 0\)

setting $z = 0$

\(\ds \)

\(=\)

\(\ds 1 + 2 \sum_{n \mathop = 1}^\infty e^{i \pi n^2 \tau}\)

Cosine of Zero is One, Power of Power

\(\ds \)

\(=\)

\(\ds 1 + 2 \sum_{n \mathop = 1}^\infty e^{i \pi n^2 \paren {i x} }\)

setting $\tau = i x$ as Definition of Jacobi Theta Function of the Third Type $x$ must be a complex constant with a positive imaginary part

\(\ds \)

\(=\)

\(\ds 1 + 2 \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}\)

$i^2 = -1$

\(\ds \leadsto \ \ \)

\(\ds \map {\vartheta_3} {0, e^{i \pi \paren {i x} } }\)

\(=\)

\(\ds 1 + 2 \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}\)

\(\ds \leadsto \ \ \)

\(\ds \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}\)

\(=\)

\(\ds \dfrac 1 2 \paren {\map {\vartheta_3} {0, e^{-\pi x } } - 1}\)

rearranging terms

$\blacksquare$