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

This category contains pages concerning Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function:

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.