The Riemann theta function is a number-theoretic function which is only really used in the derivation of the functional equation for the Riemann xi function.

The Riemann theta function is defined as:$$ \theta(x) = 2\omega (x) + 1,$$ where $\omega$ is the Riemann omega function.

The domain of the Riemann theta function is $x > 0$ .

To give an explicit form for the theta function, note that \begin{eqnarray*} \omega(x) &=& \sum_{n=1}^{\infty} e^{-n^2 \pi x}\\ &=& \sum_{n=-1}^{-\infty} e^{-(-n)^2 \pi x}\\ &=& \sum_{n=-1}^{-\infty} e^{-n^2 \pi x} \end{eqnarray*}and so \begin{eqnarray*} 2\omega(x) + 1 &=& \sum_{n=-1}^{-\infty} e^{-n^2 \pi x} + \omega(x) + 1\\ &=& \sum_{n=-1}^{-\infty} e^{-n^2 \pi x} + \sum_{n=1}^{\infty} e^{-n^2 \pi x} + e^{-0^2 \pi x}\\ &=& \sum_{n=-\infty}^{\infty} e^{-n^2 \pi x}. \end{eqnarray*}Thus we have$$ \theta(x) = \sum_{n=-\infty}^{\infty} e^{-n^2 \pi x}.$$

Riemann showed that the theta function satisfied a functional equation, which was the key step in the proof of the analytic continuation for the Riemann xi function. This has direct consequences for the Riemann zeta function.