The Fibonacci Zeta Function and Continuation

TL;DR

This paper introduces a new family of Dirichlet series related to Fibonacci numbers, generalizing the Fibonacci zeta function, and provides three methods for their meromorphic continuation to the complex plane.

Contribution

It generalizes the Fibonacci zeta function to real quadratic fields and offers three distinct methods for meromorphic continuation, including classical analytic and modular form approaches.

Findings

Defined a new family of Dirichlet series for quadratic fields

Developed three methods for meromorphic continuation

Connected Fibonacci zeta functions with modular forms

Abstract

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.