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Xi Function


\begin{figure}\begin{center}\BoxedEPSF{xiReIm.epsf scaled 700}\end{center}\end{figure}

\xi(z)\equiv {\textstyle{1\over 2}}z(z-1){\Gamma({\textstyle...
...)\Gamma({\textstyle{1\over 2}}z+1)\zeta(z)\over \sqrt{\pi^z}},
\end{displaymath} (1)

where $\zeta(z)$ is the Riemann Zeta Function and $\Gamma(z)$ is the Gamma Function (Gradshteyn and Ryzhik 1980, p. 1076). The $\xi$ function satisfies the identity
\end{displaymath} (2)

The zeros of $\xi(z)$ and of its Derivatives are all located on the Critical Strip $z=\sigma+it$, where $0<\sigma<1$. Therefore, the nontrivial zeros of the Riemann Zeta Function exactly correspond to those of $\xi(z)$. The function $\xi(z)$ is related to what Gradshteyn and Ryzhik (1980, p. 1074) call $\Xi(t)$ by
\Xi(t)\equiv \xi(z),
\end{displaymath} (3)

where $z\equiv{\textstyle{1\over 2}}+it$. This function can also be defined as
\Xi(it)\equiv{\textstyle{1\over 2}}(t^2-{\textstyle{1\over 4...
...r 2}}t+{\textstyle{1\over 4}})\zeta(t+{\textstyle{1\over 2}}),
\end{displaymath} (4)

\Xi(t)=-{\textstyle{1\over 2}}(t^2+{\textstyle{1\over 4}})\p...
...4}}-{\textstyle{1\over 2}}it)\zeta({\textstyle{1\over 2}}-it).
\end{displaymath} (5)

The de Bruijn-Newman Constant is defined in terms of the $\Xi(t)$ function.

See also de Bruijn-Newman Constant


Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, corr. enl. 4th ed. San Diego, CA: Academic Press, 1980.

© 1996-9 Eric W. Weisstein