On the von Mangoldt-type function of the Fibonacci zeta function

Abstract

[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.