How $\Xi(t)$ function in the theory of Riemann zeta function is real for real $t$?

Question

In Section 4.17. of Titchmarsh's book The Theory of the Riemann Zeta-Function it is written that for real variable $t$, the function $Z(t)$ is real. Based on the text below that should come from $\Xi(t)$ being real for $t$ real but how $\Xi(t)$ is real for $t$ real? By the functional equation for the Riemann zeta function, $\xi(\frac12+it):=\Xi(t) = \Xi(-t)$ but I cant see how this implies $\Xi(t)$ being real for $t$ real.

Answer 1

Suppose that $t\in \mathbb R$. From the definition of $\xi$, we see that it is real on the real line. Hence, by the Schwarz reflection principle, $$ \overline {\xi\! \left( {\tfrac{1}{2} + {\rm i}t} \right)} = \xi \big( {\overline {\tfrac{1}{2} + {\rm i}t} } \big) = \xi\! \left( {\tfrac{1}{2} - {\rm i}t} \right). $$ By the functional equation $\xi(1-s)=\xi(s)$, $$ \xi\! \left( {\tfrac{1}{2} - {\rm i}t} \right) = \xi\! \left( {1 - \left( {\tfrac{1}{2} + {\rm i}t} \right)} \right) = \xi\! \left( {\tfrac{1}{2} + {\rm i}t} \right). $$ Accordingly, $$ \overline {\xi\! \left( {\tfrac{1}{2} + {\rm i}t} \right)} = \xi\! \left( {\tfrac{1}{2} + {\rm i}t} \right), $$ which means that $\xi\! \left( {\tfrac{1}{2} + {\rm i}t} \right)\in \mathbb R$.