![\begin{displaymath}
\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}](x_7.gif) |
(1) |
where
is the Riemann Zeta Function and
is the Gamma Function (Gradshteyn and Ryzhik
1980, p. 1076). The
function satisfies the identity
![\begin{displaymath}
\xi(1-z)=\xi(z).
\end{displaymath}](x_11.gif) |
(2) |
The zeros of
and of its Derivatives are all located on the Critical Strip
,
where
. Therefore, the nontrivial zeros of the Riemann Zeta Function exactly correspond to those of
.
The function
is related to what Gradshteyn and Ryzhik (1980, p. 1074) call
by
![\begin{displaymath}
\Xi(t)\equiv \xi(z),
\end{displaymath}](x_16.gif) |
(3) |
where
. This function can also be defined as
![\begin{displaymath}
\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}](x_18.gif) |
(4) |
giving
![\begin{displaymath}
\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}](x_19.gif) |
(5) |
The de Bruijn-Newman Constant is defined in terms of the
function.
See also de Bruijn-Newman Constant
References
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
1999-05-20