A Riemann-type duality of shuffle Hopf algebras related to multiple zeta values
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866909878914646016 |
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| author | Guo, Li Xiang, Hongyu Zhang, Bin |
| author_facet | Guo, Li Xiang, Hongyu Zhang, Bin |
| contents | This paper offers a Hopf algebraic interpretation of a functional equation of multiple zeta functions, motivated by the classical symmetry of the Riemann zeta function. Starting from the extended shuffle algebra that encodes multiple zeta values (MZVs) at integer arguments, we show that its subalgebra corresponding to nonpositive arguments carries a natural differential Hopf algebra structure. This Hopf algebra is in graded linear duality with the shuffle Hopf algebra associated to MZVs at positive arguments. The resulting duality, realized through an explicit isomorphism, provides an algebraic analog of the functional equation relating $ζ(s)$ with $ζ(1-s)$ of the Riemann zeta function and unifies the positive and nonpositive sectors of multiple zeta functions within a common Hopf algebraic framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20952 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Riemann-type duality of shuffle Hopf algebras related to multiple zeta values Guo, Li Xiang, Hongyu Zhang, Bin Rings and Algebras 11M32, 16T05, 12H05, 16T30, 16W25, 16S10, 40B05 This paper offers a Hopf algebraic interpretation of a functional equation of multiple zeta functions, motivated by the classical symmetry of the Riemann zeta function. Starting from the extended shuffle algebra that encodes multiple zeta values (MZVs) at integer arguments, we show that its subalgebra corresponding to nonpositive arguments carries a natural differential Hopf algebra structure. This Hopf algebra is in graded linear duality with the shuffle Hopf algebra associated to MZVs at positive arguments. The resulting duality, realized through an explicit isomorphism, provides an algebraic analog of the functional equation relating $ζ(s)$ with $ζ(1-s)$ of the Riemann zeta function and unifies the positive and nonpositive sectors of multiple zeta functions within a common Hopf algebraic framework. |
| title | A Riemann-type duality of shuffle Hopf algebras related to multiple zeta values |
| topic | Rings and Algebras 11M32, 16T05, 12H05, 16T30, 16W25, 16S10, 40B05 |
| url | https://arxiv.org/abs/2412.20952 |