On the sign of the real part of the Riemann zeta-function

We consider the distribution of $\arg \zeta (\sigma +it)$ on fixed lines $\sigma >\frac{1}{2}$, and in particular the density $$d(\sigma )=\underset{T\to +\mathrm{\infty }}{\lim }\frac{1}{2T}\mathrm{\mid }\{t\in [-T,+T]:\mathrm{\mid }\arg \zeta (\sigma +it)\mathrm{\mid }>\pi \mathrm{/}2\}\mathrm{\mid },$$ and the closely related density $$d_{-}(\sigma )=\underset{T\to +\mathrm{\infty }}{\lim }\frac{1}{2T}\mathrm{\mid }\{t\in [-T,+T]:\mathrm{\Re }\zeta (\sigma +it)<0\}\mathrm{\mid }\mathrm{.}$$ Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function ${\psi }_{\sigma }(x)$ associated with $\arg \zeta (\sigma +it)$. We give explicit expressions for $d(\sigma )$ and $d_{-}(\sigma )$ in terms of ${\psi }_{\sigma }(x)$. Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of $d(\sigma )$ and $d_{-}(\sigma )$.