  We prove that if $\delta>0$ and $p$ is real then $$ \sup_T \int_T^{T+\delta}
|\zeta(1+it)|^p dt <\infty,$$ if and only if $-1<p<1$. Furthermore, we show the
omega estimates $$
  \int_T^{T+\delta} |\zeta(1+it)|^{\pm 1} dt = \Omega(\log \log \log T),$$ $$
\int_T^{T+\delta} |\zeta(1+it)|^{\pm p} dt = \Omega((\log \log T)^{p-1}),
\qquad (p>1) $$ which with the exception of an additional $\log \log \log T$
factor in the second estimate coincides with conditional (under the Riemann
hypothesis) order estimates. We also prove weaker unconditional order
estimates.
