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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 $t$, the Riemann-Siegel $Z$ 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 $Z(t)$ (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.




© 1996-9 Eric W. Weisstein
1999-05-25