Xi Function

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

$$\xi(1-z)=\xi(z).$$ (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

$$\Xi(t)\equiv \xi(z),$$ (3)

where $z\equiv{\textstyle{1\over 2}}+it$. This function can also be defined as

$$\Xi(it)\equiv{\textstyle{1\over 2}}(t^2-{\textstyle{1\over 4... ...r 2}}t+{\textstyle{1\over 4}})\zeta(t+{\textstyle{1\over 2}}),$$ (4)

giving

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

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