PlanetMath: Riemann $þeta$-function

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Riemann

-function

(Definition)

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:

$\begin{displaymath} \theta(x) = 2\omega (x) + 1, \end{displaymath}$

where $\omega$ is the Riemann omega function.

The domain of the Riemann theta function is .

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

$\begin{eqnarray*} \omega(x) &=& \sum_{n=1}^{\infty} e^{-n^2 \pi x}\ &=& \sum_{... ...y} e^{-(-n)^2 \pi x}\ &=& \sum_{n=-1}^{-\infty} e^{-n^2 \pi x} \end{eqnarray*}$

and so

$\begin{eqnarray*} 2\omega(x) + 1 &=& \sum_{n=-1}^{-\infty} e^{-n^2 \pi x} + \ome... ... e^{-0^2 \pi x}\ &=& \sum_{n=-\infty}^{\infty} e^{-n^2 \pi x}. \end{eqnarray*}$

Thus we have

$\begin{displaymath} \theta(x) = \sum_{n=-\infty}^{\infty} e^{-n^2 \pi x}. \end{displaymath}$

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.

"Riemann

-function" is owned by PrimeFan. [ full author list (2) | owner history (3) ]

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See Also: Landsberg-Schaar relation

Other names:

Riemann theta-function, Riemann theta function

This object's parent.

Attachments:: functional equation for the theta function (Theorem) by rspuzio

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Cross-references: Riemann zeta function, consequences, Riemann Xi function, analytic continuation, functional equation, functional equation for the Riemann Xi function, derivation, function
There is 1 reference to this entry.

This is version 10 of Riemann $þeta$

-function, born on 2003-01-29, modified 2006-10-13.
Object id is 3940, canonical name is RiemannThetaFunction.
Accessed 2961 times total.

Classification:

AMS MSC:

11M06 (Number theory :: Zeta and $L$-functions: analytic theory :: $\zeta $)

Pending Errata and Addenda

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