Correlations of the Riemann zeta function
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910447143223296 |
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| author | Curran, Michael J. |
| author_facet | Curran, Michael J. |
| contents | Assuming the Riemann hypothesis, we investigate the shifted moments of the zeta function \[ M_{α,β}(T) = \int_T^{2T} \prod_{k = 1}^m |ζ(\tfrac{1}{2} + i (t + α_k))|^{2 β_k} dt \] introduced by Chandee, where $α = α(T) = (α_1, \ldots, α_m)$ and $β = (β_1 \ldots , β_m)$ satisfy $|α_k| \leq T/2$ and $β_k\geq 0$. We shall prove that \[ M_{α,β}(T) \ll_{β} T (\log T)^{β_1^2 + \cdots + β_m^2} \prod_{1\leq j < k \leq m} |ζ(1 + i(α_j - α_k) + 1/ \log T )|^{2β_j β_k}. \] This improves upon the previous best known bounds due to Chandee and Ng, Shen, and Wong, particularly when the differences $|α_j - α_k|$ are unbounded as $T \rightarrow \infty$. The key insight is to combine work of Heap, Radziwiłł, and Soundararajan and work of the author with the work of Harper on the moments of the zeta function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_10123 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Correlations of the Riemann zeta function Curran, Michael J. Number Theory Assuming the Riemann hypothesis, we investigate the shifted moments of the zeta function \[ M_{α,β}(T) = \int_T^{2T} \prod_{k = 1}^m |ζ(\tfrac{1}{2} + i (t + α_k))|^{2 β_k} dt \] introduced by Chandee, where $α = α(T) = (α_1, \ldots, α_m)$ and $β = (β_1 \ldots , β_m)$ satisfy $|α_k| \leq T/2$ and $β_k\geq 0$. We shall prove that \[ M_{α,β}(T) \ll_{β} T (\log T)^{β_1^2 + \cdots + β_m^2} \prod_{1\leq j < k \leq m} |ζ(1 + i(α_j - α_k) + 1/ \log T )|^{2β_j β_k}. \] This improves upon the previous best known bounds due to Chandee and Ng, Shen, and Wong, particularly when the differences $|α_j - α_k|$ are unbounded as $T \rightarrow \infty$. The key insight is to combine work of Heap, Radziwiłł, and Soundararajan and work of the author with the work of Harper on the moments of the zeta function. |
| title | Correlations of the Riemann zeta function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2303.10123 |