# Riemann $\theta$-function

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 (http://planetmath.org/Function) of the Riemann theta function is $x>0$.

To give an explicit form for the theta function, note that

 $\displaystyle\omega(x)$ $\displaystyle=$ $\displaystyle\sum_{n=1}^{\infty}e^{-n^{2}\pi x}$ $\displaystyle=$ $\displaystyle\sum_{n=-1}^{-\infty}e^{-(-n)^{2}\pi x}$ $\displaystyle=$ $\displaystyle\sum_{n=-1}^{-\infty}e^{-n^{2}\pi x}$

and so

 $\displaystyle 2\omega(x)+1$ $\displaystyle=$ $\displaystyle\sum_{n=-1}^{-\infty}e^{-n^{2}\pi x}+\omega(x)+1$ $\displaystyle=$ $\displaystyle\sum_{n=-1}^{-\infty}e^{-n^{2}\pi x}+\sum_{n=1}^{\infty}e^{-n^{2}% \pi x}+e^{-0^{2}\pi x}$ $\displaystyle=$ $\displaystyle\sum_{n=-\infty}^{\infty}e^{-n^{2}\pi x}.$

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.

Title Riemann $\theta$-function Riemannthetafunction 2013-03-22 13:23:58 2013-03-22 13:23:58 PrimeFan (13766) PrimeFan (13766) 13 PrimeFan (13766) Definition msc 11M06 Riemann theta-function Riemann theta function LandsbergSchaarRelation