The Fibonacci Zeta Function and Continuation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916607912050688 |
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| author | Assaf, Eran Kuan, Chan Ieong Lowry-Duda, David Walker, Alexander |
| author_facet | Assaf, Eran Kuan, Chan Ieong Lowry-Duda, David Walker, Alexander |
| contents | We introduce a family of Dirichlet series associated to real quadratic number fields that generalize the ordinary Fibonacci zeta function $\sum F(n)^{-s}$, where $F(n)$ denotes the $n$th Fibonacci number. We then give three different methods of meromorphic continuation to $\mathbb{C}$. Two are purely analytic and classical, while the third uses shifted convolutions and modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13620 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Fibonacci Zeta Function and Continuation Assaf, Eran Kuan, Chan Ieong Lowry-Duda, David Walker, Alexander Number Theory 11B39, 11M41, 30B40 We introduce a family of Dirichlet series associated to real quadratic number fields that generalize the ordinary Fibonacci zeta function $\sum F(n)^{-s}$, where $F(n)$ denotes the $n$th Fibonacci number. We then give three different methods of meromorphic continuation to $\mathbb{C}$. Two are purely analytic and classical, while the third uses shifted convolutions and modular forms. |
| title | The Fibonacci Zeta Function and Continuation |
| topic | Number Theory 11B39, 11M41, 30B40 |
| url | https://arxiv.org/abs/2412.13620 |