On page 119 Edwards rewrites the $\xi(s)$ function as
$$ \begin{align*} \xi\left (\frac{1}{2} + it \right ) &= \frac{s}{2}\Pi\left ( \frac{s}{2} - 1 \right )\cdot \pi^{-s/2}\cdot (s - 1) \cdot \zeta(s)\\\\ \\\\ &= e^{\ln \Pi[(s/2) - 1]}\cdot \pi^{-s/2} \cdot \frac{s(s-1)}{2}\cdot \zeta(s)\\\\ \\\\ &= \left [ e^{Re \ln \Pi[(s/2)-1]} \cdot \pi^{-1/4}\cdot \frac{-t^2 - \frac{1}{4}}{2} \right ]\\\\ \\\\ &\times \left [ e^{i Im \ln \Pi[(s/2) - 1]} \cdot \pi^{-it/2} \cdot \zeta \left ( \frac{1}{2} + it \right )\right ] \end{align*} $$
(where $s = 1/2 + it$)
Can you explain how he obtains the expressions inside the brackets?
He seems to separate the real and imaginary parts but then he has a product not addition.
eterms but I don't get the initialeterm outside the brackets. Why $\zeta$ is in the second bracket? Is there a name of what he is doing here so that I can look it up further? – zeynel Mar 31 '24 at 11:26