Suppose analytic continuation of the Riemann zeta function $\zeta(z)$, defined for all $z \in \mathbb C$, and let $x \in \mathbb R$.
Then, $\Big| \frac{\zeta(1/2 + ix)}{\zeta(1/2 - ix)}\Big| \equiv 1$? How would one show this identity?
2026-04-07 04:47:55.1775537275
Is $\Big| \frac{\zeta(1/2 + ix)}{\zeta(1/2 - ix)} \Big| \equiv 1$ for all $x \in \mathbb R$?
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The $\zeta$ function is real for real parameters, hence $$ \forall w\in\mathbb{C},\qquad \zeta(\overline{w}) = \overline{\zeta(w)} $$ implying the statement, follows from Schwarz reflection principle.