An application of functional analysis to the Riemann zeta function

Abstract

Lindel\"of conjectured that the Riemann zeta function $\zeta(\sigma+it)$ grows more slowly than any fixed positive power of $t$ as $t\rightarrow\infty$ when $\sigma\geq 1/2$. Hardy and Littlewood showed that this is equivalent to the existence of the $2k$th moments for all fixed $k\in\mathbb{N}$ and $\sigma>1/2$. 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 $\sigma>\sigma_k>1/2$ then it also exists on the line $\sigma=\sigma_k$.