An application of functional analysis to the Riemann zeta function
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916626342871040 |
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| author | Smith, Kevin |
| author_facet | Smith, Kevin |
| contents | 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 $σ=σ_k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01376 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An application of functional analysis to the Riemann zeta function Smith, Kevin Number Theory 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 $σ=σ_k$. |
| title | An application of functional analysis to the Riemann zeta function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2312.01376 |