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

TL;DR

This paper investigates the growth behavior of the $L^p$ norms of the Riemann zeta-function along the line Re$(s)=1$, establishing finiteness conditions and omega estimates that match conditional bounds under the Riemann hypothesis.

Contribution

It provides new necessary and sufficient conditions for the boundedness of $L^p$ norms of $ ext{zeta}(1+it)$ and derives omega estimates that align with conditional order estimates.

Findings

Finite $L^p$ norms for $-1<p<1$ on short intervals.

Omega estimates for $| ext{zeta}(1+it)|^{ ext{power}}$ growth.

Conditional bounds match the derived omega estimates.

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.