On the growth of the $L^p$ norm of the Riemann zeta-function on the line Re$(s)=1$

Abstract

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.