We study Omega theorems for the expression $E=\text{Re}(e^{i\theta} \log \zeta(\sigma_0+it_0))$ where $1/2\le\sigma_0<1$ and $0\le\theta<2\pi$ ($\sigma_0$, $\theta$ fixed) as $t_0\to\infty$. In fact we prove $E\ge C(1-\sigma_0)^{-1} (\log t_0)^{1-\sigma_0}(\log\log t_0)^{-\sigma_0}$ for at least one $t_0$ in $[T^{\varepsilon},T]$ where $C$ is a positive constant. Note that $(1-\sigma_0)^{-1}\to\bb$ as $\sigma_0\to1$.