# Riemann $\varpi$ function

The Riemann $\varpi$ function is used in the proof of the analytic continuation for the Riemann Xi function to the whole complex plane. It is defined as:

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

This function is a special case of a Jacobi $\vartheta$ function (http://planetmath.org/JacobiVarthetaFunctions):

 $\varpi(x)=\vartheta_{3}(0|ix)$

As such the $\varpi$ function satisfies a functional equation, which a special case of Jacobi’s Identity for the $\vartheta$ function (http://planetmath.org/JacobisIdentityForVarthetaFunctions).

Title Riemann $\varpi$ function RiemannvarpiFunction 2013-03-22 13:24:12 2013-03-22 13:24:12 rspuzio (6075) rspuzio (6075) 12 rspuzio (6075) Definition msc 11M06