Functional equation, upper bounds and analogue of Lindelöf hypothesis for the Barnes double zeta function
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915996274524160 |
|---|---|
| author | Miyagawa, Takashi |
| author_facet | Miyagawa, Takashi |
| contents | The functional equations of the Riemann zeta function, the Hurwitz zeta function, and the Lerch zeta function have been well known for a long time, and there is great importance in studying these zeta functions. For example, fundamental properties such as the upper bounds, the distribution of zeros, and the zero-free regions in the Riemann zeta function derive from functional equations.
In this paper, we consider the functional equations for the Barnes double zeta-function $ ζ_2 (s, α; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (α+vm+wn)^{-s} $. Additionally, by applying this functional equation and the Phragmén-Lindelöf convexity principle, we obtain some upper bounds for $ ζ_2(σ+ it, α; v, w) $ with respect to $ t $ as $ t \rightarrow \infty $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_10786 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Functional equation, upper bounds and analogue of Lindelöf hypothesis for the Barnes double zeta function Miyagawa, Takashi Number Theory Primary 11M32, Secondary 11B35 The functional equations of the Riemann zeta function, the Hurwitz zeta function, and the Lerch zeta function have been well known for a long time, and there is great importance in studying these zeta functions. For example, fundamental properties such as the upper bounds, the distribution of zeros, and the zero-free regions in the Riemann zeta function derive from functional equations. In this paper, we consider the functional equations for the Barnes double zeta-function $ ζ_2 (s, α; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (α+vm+wn)^{-s} $. Additionally, by applying this functional equation and the Phragmén-Lindelöf convexity principle, we obtain some upper bounds for $ ζ_2(σ+ it, α; v, w) $ with respect to $ t $ as $ t \rightarrow \infty $. |
| title | Functional equation, upper bounds and analogue of Lindelöf hypothesis for the Barnes double zeta function |
| topic | Number Theory Primary 11M32, Secondary 11B35 |
| url | https://arxiv.org/abs/2208.10786 |