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.