A note on the real part of the Riemann zeta-function

We consider the real part $\mathrm{\Re }(\zeta (s))$ of the Riemann zeta-function $\zeta (s)$ in the half-plane $\mathrm{\Re }(s)\ge 1$. We show how to compute accurately the constant ${\sigma }_0=1.19\dots$ which is defined to be the supremum of $\sigma$ such that $\mathrm{\Re }(\zeta (\sigma +it))$ can be negative (or zero) for some real $t$. We also consider intervals where $\mathrm{\Re }(\zeta (1+it))\le 0$ and show that they are rare. The first occurs for $t$ approximately 682112.9, and has length about 0.05. We list the first fifty such intervals.