On single-variable Witten zeta functions of rank two and three
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911265054523392 |
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| author | Au, Kam Cheong |
| author_facet | Au, Kam Cheong |
| contents | By introducing a novel integration kernel for Mellin transform, we uncover many previously unknown and intriguing properties of the Witten zeta functions of rank two and three. Detailed results concerning their pole locations, residues, and special values are obtained. We propose a non-trivial conjecture regarding their derivatives at the origin, which seems to encode deep information about the root system. We also discuss their behavior at negative integers, highlighting a connection with Eisenstein series and a $p$-adic observation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17196 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On single-variable Witten zeta functions of rank two and three Au, Kam Cheong Number Theory Classical Analysis and ODEs 11M32, 11M35 (Primary) 11P82, 33C220 (Secondary) By introducing a novel integration kernel for Mellin transform, we uncover many previously unknown and intriguing properties of the Witten zeta functions of rank two and three. Detailed results concerning their pole locations, residues, and special values are obtained. We propose a non-trivial conjecture regarding their derivatives at the origin, which seems to encode deep information about the root system. We also discuss their behavior at negative integers, highlighting a connection with Eisenstein series and a $p$-adic observation. |
| title | On single-variable Witten zeta functions of rank two and three |
| topic | Number Theory Classical Analysis and ODEs 11M32, 11M35 (Primary) 11P82, 33C220 (Secondary) |
| url | https://arxiv.org/abs/2412.17196 |