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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.10888 |
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| _version_ | 1866913354434478080 |
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| author | Heap, Winston Sahay, Anurag |
| author_facet | Heap, Winston Sahay, Anurag |
| contents | We prove a sharp upper bound for the fourth moment of the Hurwitz zeta function $ζ(s,α)$ on the critical line when the shift parameter $α$ is irrational and of irrationality exponent strictly less than 3. As a consequence, we determine the order of magnitude of the $2k$th moment for all $0 \leqslant k \leqslant 2$ in this case. In contrast to the Riemann zeta function and other $L$-functions from arithmetic, these grow like $T (\log T)^k$. This suggests, and we conjecture, that the value distribution of $ζ(s,α)$ on the critical line is Gaussian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10888 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The fourth moment of the Hurwitz zeta function Heap, Winston Sahay, Anurag Number Theory We prove a sharp upper bound for the fourth moment of the Hurwitz zeta function $ζ(s,α)$ on the critical line when the shift parameter $α$ is irrational and of irrationality exponent strictly less than 3. As a consequence, we determine the order of magnitude of the $2k$th moment for all $0 \leqslant k \leqslant 2$ in this case. In contrast to the Riemann zeta function and other $L$-functions from arithmetic, these grow like $T (\log T)^k$. This suggests, and we conjecture, that the value distribution of $ζ(s,α)$ on the critical line is Gaussian. |
| title | The fourth moment of the Hurwitz zeta function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.10888 |