Riemann zeta function $\zeta(s)$ is related to Riemann Xi function $\Xi(z)$ via: $$s=\frac12+ iz,\qquad \Xi(z):=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\tag{1}$$
We found the following function $\Xi_2(t)$ to approximate $\Xi(z)$ on the critical line ($Im(z)=0, Re(z)=t>=12$):
\[CapitalXi]2[t_] := Sum[(k^2*Pi)^(-(1/4) + (I*t)/2)*
((-(1/4) - (I*t)/2)*Gamma[5/4 - (I*t)/2, (k*Pi)^2/t] +
((k*Pi)^2/t)^(5/4 - (I*t)/2)*Exp[-((k*Pi)^2/t)]) +
(k^2*Pi)^(-(1/4) - (I*t)/2)*((-(1/4) + (I*t)/2)*
Gamma[5/4 + (I*t)/2, (k*Pi)^2/t] + ((k*Pi)^2/t)^(5/4 + (I*t)/2)*
Exp[-((k*Pi)^2/t)]), {k, 1, t/Pi}]
Let $M(t)=t^2\exp(-\pi t/4)$ be a normalization function:
M[t_] := t^2*Exp[-(t*(Pi/4))]
Here are the two plots of $\Xi(t),\Xi_2(t)$ vs. $t$ at different $t$. Green color is for $\Xi(t)$ and Red color is for $\Xi_2(t)$.
You can clearly see that $\Xi_2(t)$ does not behave properly beyond $t=180$.
Question: how can we plot $\Xi_2(t)$ beyond $t=180$?
Plot[-RiemannSiegelZ[x], {x, 150, 200}]
. $\endgroup$ – Mariusz Iwaniuk May 31 '20 at 9:18