Analogues of the Lindelöf Hypothesis for the Barnes multiple zeta function and related problems
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917556150861824 |
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| author | Miyagawa, Takashi |
| author_facet | Miyagawa, Takashi |
| contents | For the Lindelöf Hypothesis concerning the Riemann zeta function $ζ(s)$, upper bounds as $\Im(s)\to\infty$ have been extensively studied for many years. In particular, the Lindelöf Hypothesis is one of the most important open problems in analytic number theory. It is also known to be equivalent to certain mean value estimates, which provide a fundamental connection between pointwise upper bounds and integral mean values of zeta-functions.
In this paper, we consider an analogue of the Lindelöf Hypothesis for the Barnes multiple zeta function $ζ_r (s,a,(w_1,\dots,w_r)) = \sum_{m_1=0}^\infty \cdots \sum_{m_r=0}^\infty (a+m_1 w_1+\cdots+m_r w_r)^{-s} $, and establish equivalent conditions in terms of integral mean values. In particular, the situation depends essentially on the $\Q$-rank of $\langle w_1,\dots,w_r\rangle$, and it is especially interesting that phenomena peculiar to the Barnes multiple zeta function appear according to this rank. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02407 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analogues of the Lindelöf Hypothesis for the Barnes multiple zeta function and related problems Miyagawa, Takashi Number Theory 11M32, 11B35 For the Lindelöf Hypothesis concerning the Riemann zeta function $ζ(s)$, upper bounds as $\Im(s)\to\infty$ have been extensively studied for many years. In particular, the Lindelöf Hypothesis is one of the most important open problems in analytic number theory. It is also known to be equivalent to certain mean value estimates, which provide a fundamental connection between pointwise upper bounds and integral mean values of zeta-functions. In this paper, we consider an analogue of the Lindelöf Hypothesis for the Barnes multiple zeta function $ζ_r (s,a,(w_1,\dots,w_r)) = \sum_{m_1=0}^\infty \cdots \sum_{m_r=0}^\infty (a+m_1 w_1+\cdots+m_r w_r)^{-s} $, and establish equivalent conditions in terms of integral mean values. In particular, the situation depends essentially on the $\Q$-rank of $\langle w_1,\dots,w_r\rangle$, and it is especially interesting that phenomena peculiar to the Barnes multiple zeta function appear according to this rank. |
| title | Analogues of the Lindelöf Hypothesis for the Barnes multiple zeta function and related problems |
| topic | Number Theory 11M32, 11B35 |
| url | https://arxiv.org/abs/2606.02407 |