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An application of functional analysis to the Riemann zeta function
- Publication Year :
- 2023
-
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 $\sigma=\sigma_k$.<br />Comment: 10 pages
- Subjects :
- Mathematics - Number Theory
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2312.01376
- Document Type :
- Working Paper