Reflection symmetry of $\zeta$: $\zeta(\bar{s}) = \overline{\zeta(s)}$

For every complex number $s$, the Riemann zeta function commutes with complex conjugation:

This is the Schwarz reflection property of $\zeta$, valid on all of $\mathbb{C}$ (with the completed meromorphic continuation): it holds because $\zeta$ is real on the real axis where its Dirichlet series converges, and the identity propagates to the full plane by analytic continuation.

The reflection identity halves the work in zero-free-region and growth estimates: any bound on $\zeta$ or on $|\zeta(\sigma + it)|$ established for $t \ge 0$ transfers immediately to $t \le 0$, and zeros of $\zeta$ come in conjugate pairs. The PNT+ development uses it to reduce vertical-strip estimates to the upper half-plane.