An application of functional analysis to the Riemann zeta function

Abstract

Lindelöf conjectured that the Riemann zeta function $\hat{I}\P (\ddot{I}ƒ+it)$ grows more slowly than any fixed positive power of $t$ as $t\to \infty$ when $\ddot{I}ƒ\ge 1/2$. Hardy and Littlewood showed that this is equivalent to the existence of the $2k$th moments for all fixed $k\in \mathbb{N}$ and $\ddot{I}ƒ>1/2$. 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 $\ddot{I}ƒ>\ddot{I}{ƒ}_k>1/2$ then it also exists on the line $\ddot{I}ƒ=\ddot{I}{ƒ}_k$.

10 pages