Any solution to $\zeta(s) = 1$ must have real part $\sigma \le \sigma(1)$ where $\sigma(1)$ is equal to the unique solution $\sigma>1$ of the equation $$\zeta(\sigma)=\frac{2^\sigma+1}{2^\sigma-1}$$ Numerically $\sigma(1)=1.9401016837\dots$
$\sigma(1)$ is the best possible constant here.
See the paper arXiv:1107.5134 where other problems of this type are considered.