On the error term of the fourth moment of the Riemann zeta-function
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916802660925440 |
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| author | Palojärvi, Neea Trudgian, Tim |
| author_facet | Palojärvi, Neea Trudgian, Tim |
| contents | We examine the size of $E_{2}(T)$, the error term in the asymptotic formula for $\int_{0}^{T} |ζ(1/2 + it)|^{4}\, dt$ where $ζ(s)$ is the Riemann zeta-function. We make improvements in the powers of $\log T$ in the known bounds for $E_{2}(T)$ and $\int_{0}^{T} E_{2}(t)^{2}\, dt$. As a consequence, we obtain small logarithmic improvements for $k$th moments where $8\leq k\leq 12$. In particular, we make a modest improvement on the 12th power moment for $ζ(s)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16766 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the error term of the fourth moment of the Riemann zeta-function Palojärvi, Neea Trudgian, Tim Number Theory 11M06 We examine the size of $E_{2}(T)$, the error term in the asymptotic formula for $\int_{0}^{T} |ζ(1/2 + it)|^{4}\, dt$ where $ζ(s)$ is the Riemann zeta-function. We make improvements in the powers of $\log T$ in the known bounds for $E_{2}(T)$ and $\int_{0}^{T} E_{2}(t)^{2}\, dt$. As a consequence, we obtain small logarithmic improvements for $k$th moments where $8\leq k\leq 12$. In particular, we make a modest improvement on the 12th power moment for $ζ(s)$. |
| title | On the error term of the fourth moment of the Riemann zeta-function |
| topic | Number Theory 11M06 |
| url | https://arxiv.org/abs/2506.16766 |