Riemann-Siegel Functions

$\begin{figure}\begin{center}\BoxedEPSF{RiemannSiegelZ.epsf}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{RiemannSiegelZReIm.epsf scaled 750}\end{center}\end{figure}$

For a Real Positive , the Riemann-Siegel function is defined by

$\begin{displaymath} Z(t)\equiv e^{i\vartheta(t)}\zeta({\textstyle{1\over 2}}+it). \end{displaymath}$

The top plot superposes

(thick line) on $\vert\zeta({\textstyle{1\over 2}}+it)\vert$ , where $\zeta(z)$ is the Riemann Zeta Function.

$\begin{figure}\begin{center}\BoxedEPSF{RiemannSiegelTheta.epsf}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{RiemannSiegelThetaReIm.epsf scaled 750}\end{center}\end{figure}$

The Riemann-Siegel theta function appearing above is defined by

$\displaystyle \vartheta$	$\textstyle \equiv$	$\displaystyle \Im[\ln\Gamma({\textstyle{1\over 4}}+{\textstyle{1\over 2}}it)-{\textstyle{1\over 2}}t\ln \pi]$
	$\textstyle =$	$\displaystyle \arg[\Gamma({\textstyle{1\over 4}}+{\textstyle{1\over 2}}it)]-{\textstyle{1\over 2}}t\ln \pi.$

These functions are implemented in Mathematica ${}^{\scriptstyle\circledRsymbol}$
(Wolfram Research, Champaign, IL) as RiemannSiegelZ[z] and RiemannSiegelTheta[z], illustrated above.

See also Riemann Zeta Function

References

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 143, 1991.