Riemann θ-function

The Riemann theta functionDlmfDlmfMathworldPlanetmath 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:


where ω is the Riemann omega function.

The domain (http://planetmath.org/FunctionMathworldPlanetmath) of the Riemann theta function is x>0.

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

ω(x) = n=1e-n2πx
= n=-1-e-(-n)2πx
= n=-1-e-n2πx

and so

2ω(x)+1 = n=-1-e-n2πx+ω(x)+1
= n=-1-e-n2πx+n=1e-n2πx+e-02πx
= n=-e-n2πx.

Thus we have


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 functionDlmfDlmfMathworldPlanetmath.

Title Riemann θ-function
Canonical name Riemannthetafunction
Date of creation 2013-03-22 13:23:58
Last modified on 2013-03-22 13:23:58
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 13
Author PrimeFan (13766)
Entry type Definition
Classification msc 11M06
Synonym Riemann theta-function
Synonym Riemann theta function
Related topic LandsbergSchaarRelation