Reflection symmetry of the Riemann zeta function: $\overline{\zeta(\overline{s})} = \zeta(s)$

For every complex number $s$, the Riemann zeta function satisfies the conjugation symmetry

where $\overline{z}$ denotes complex conjugation. Equivalently, $\zeta$ commutes with reflection in the real axis: the value of $\zeta$ at the mirror point $\overline{s}$ is the mirror of the value at $s$.

For $\operatorname{Re}(s) > 1$ this is immediate from the Dirichlet series $\zeta(s) = \sum n^{-s}$, whose coefficients are real; the full statement extends the symmetry to all of $\mathbb{C}$ (including the continuation past the pole at $s = 1$) by the Schwarz reflection principle / uniqueness of analytic continuation. This is the standard "reality" property of $\zeta$: it forces the non-trivial zeros to come in conjugate pairs $\rho, \overline{\rho}$, and in the PNT+ project it lets bounds and zero-free-region statements proved for $t > 0$ be transferred automatically to $t < 0$.