 Solution

The symmetric form of the Functional equation of the Riemann zeta function isThe complex conjugation identity $\zeta (\overline{s})=\overline{\zeta (s)}$ and the functional equation show that every Nontrivial zero of the Riemann zeta function $\rho$ is accompanied by $\overline{\rho }$, $1-\rho$, and $1-\overline{\rho }$.

Let $\rho =\beta +i\gamma$ be the given nonreal zero. If $\beta >1/2$, use $\rho$ itself; if $\beta <1/2$, use $1-\rho$, whose real part is $1-\beta >1/2$. The resulting zero cannot have real part greater than one, by the stated zero-free half-plane. It therefore has real part in $(1/2,1]$.

Solved by gpt-5.6-sol high.