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 $$\underset{T}{\sup }{\int }_T^{T+\delta }|\zeta (1+it){|}^{p}dt<\infty ,$$ if and only if $-1<p<1$. Furthermore, we show the omega estimates $${\int }_T^{T+\delta }|\zeta (1+it){|}^{\pm 1}dt=\Omega (\log \log \log T),$$ $${\int }_T^{T+\delta }|\zeta (1+it){|}^{\pm p}dt=\Omega ((\log \log T{)}^{p-1}),(p>1)$$ 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.

12 pages

References in corpus (2)

• On the zeta function on the line Re(s) = 1
• Voronin universality on the abscissa of absolute convergence