The Dirichlet series associated to the Fibonacci sequence \(\{F_{n}\}\) , \(\begin{aligned} \sum _{n=1}^{\infty } F_{n}^{-s}, \end{aligned}\) converges for \(s\in \mathbb {C}\) with \(\operatorname {Re}s > 0\) . 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 \(\begin{aligned} -\sum _{l=1}^{\infty } \Lambda _{\mathcal {F}}(l)l^{-s}, \end{aligned}\) 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.
