An application of functional analysis to the Riemann zeta function

TL;DR

The paper connects the Lindelöf conjecture for the Riemann zeta function to the completeness of the Besicovitch space $B^2$ of almost-periodic functions. By proving that $\zeta(\sigma+it)^k$ lies in $B^2$ for $\sigma>\sigma_k$ and constructing a $B^2$-limit as $\sigma\downarrow\sigma_k$, it derives that $\zeta^k(\sigma_k+it)$ also lies in $B^2$; Abelian–Tauberian arguments then yield the boundary moment asymptotic $\frac{1}{T}\int_{1}^{T}|\zeta(\sigma_k+it)|^{2k}dt \to \sum_{n\ge1} d_k(n)^2/n^{2\sigma_k}$. A key ingredient is the notion of concentration on null sets and the associated Proposition CNS, which ties $B^2$-membership to moment behavior and bounded perturbations. Together, these results provide a functional-analytic route to extend Hardy–Littlewood–Ingham type moment identities to the critical boundary line and illuminate a path toward understanding Lindelöf via Hilbert-space properties of Besicovitch almost-periodic functions.

Abstract

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

Theorems & Definitions (7)

• Theorem 1
• Lemma 1
• Lemma 2
• Proposition 1
• Lemma 3
• Lemma 4
• proof