On the von Mangoldt-type function of the Fibonacci zeta function
[EN]The Dirichlet series associated to the Fibonacci sequence $\{F_{n}\}$, $$\sum_{n=1}^{\infty} F_{n}^{-s},$$ converges for $s\in \mathbb{C}$ with $\Re s > 0$. The analytic function $\varphi(s)$ it defines on the right half-plane is known as the Fibonacci zeta function. Here we consider its loga...
| Autores: | , , |
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| Tipo de documento: | artigo |
| Estado: | Versão publicada |
| Data de publicação: | 2025 |
| País: | España |
| Recursos: | Universidad de Salamanca (USAL) |
| Repositório: | GREDOS. Repositorio Institucional de la Universidad de Salamanca |
| OAI Identifier: | oai:gredos.usal.es:10366/168506 |
| Acesso em linha: | http://hdl.handle.net/10366/168506 |
| Access Level: | Acesso embargado |
| Palavra-chave: | Fibonacci numbers Fibonacci zeta function Dirichlet series Von Mangoldt Fibonacci function. 12 Matemáticas |
| Resumo: | [EN]The Dirichlet series associated to the Fibonacci sequence $\{F_{n}\}$, $$\sum_{n=1}^{\infty} F_{n}^{-s},$$ converges for $s\in \mathbb{C}$ with $\Re s > 0$. The analytic function $\varphi(s)$ it defines on the right half-plane is known as the Fibonacci zeta function. Here we consider its logarithmic derivative $\varphi'(s)/\varphi(s)$, which formally corresponds to the Dirichlet series $$-\sum_{l=1}^{\infty} \Lambda_{\mathcal{F}}(l) l^{-s},$$ where the arithmetical function $\Lambda_{\mathcal{F}}(l)$ can be considered analogous to the classical von Mangoldt function $\Lambda(s)$, which is defined by $\zeta'(s)/\zeta(s) = -\sum_{n=1}^{\infty} \Lambda(n) n^{-s}$ where $\zeta(s)$ is the Riemann zeta function. This paper studies some properties of the function $\Lambda_{\mathcal{F}}(l)$ along with the domain of convergence of this Dirichlet series. |
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