According to Wikipedia, there is a global convergent series for Riemann Zeta function:
https://en.wikipedia.org/wiki/Riemann_zeta_function#Globally_convergent_series
Is there a similar global convergent series for Riemann Xi function ?
2026-04-05 03:16:05.1775358965
is there a Globally convergent series for Riemann Xi function?
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Riemann Xi function can be defined via Riemann zeta function as:
$$\Xi(z)=\xi(1/2+iz)$$ $$\xi(s)=(1/2)s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$
Since $\Xi(z)$ is an entire function of $z$, it can be Taylor-expanded at any point $z_0(\not=\infty)$ in the complex plane.
You can find out more in the following references:
(1)Riemann's Zeta FunctionJun 13, 2001 by Harold M. Edwards
(2)The Theory of the Riemann Zeta-Function (Oxford Science Publications)Feb 5, 1987 by E. C. Titchmarsh and D. R. Heath-Brown