An application of functional analysis to the Riemann zeta function

Lindelöf conjectured that the Riemann zeta function $\zeta (\sigma +it)$ grows more slowly than any fixed positive power of $t$ as $t\to \mathrm{\infty }$ when $\sigma \ge 1\mathrm{/}2$. Hardy and Littlewood showed that this is equivalent to the existence of the $2k$th moments for all fixed $k\in \mathbb{N}$ and $\sigma >1\mathrm{/}2$. In this paper we show that the completeness of the Hilbert space $B^2$ of Besicovitch almost-periodic functions implies that if the $2k$th moment exists for $\sigma >{\sigma }_k>1\mathrm{/}2$ then it also exists on the line $\sigma ={\sigma }_k$.