The Fibonacci Zeta Function

We consider the lacunary Dirichlet series obtained by taking the reciprocals of the s-th powers of the Fibonacci numbers. This series admits an analytic continuation to the entire complex plane. Its special values at integral arguments are then studied. If the argument is a negative integer, the value is algebraic. If the argument is a positive even integer, the value is transcendental by Nesterenko’s work. This is a result of Duverney, Ke. Nishioka, Ku. Nishioka and Shiokawa. We present a simplified proof of their result. If the argument is 1, the value has been shown to be irrational by André-Jeannin. We present a slight modification of Duverney’s proof of this fact. At the same time, we highlight the “modular connection” of these questions as well as signal some new results in the theory of special values of q-analogues of classical Dirichlet L-functions.