The Fibonacci Zeta Function and Modular Forms

TL;DR

This paper introduces a new class of Dirichlet series extending the Fibonacci zeta function and demonstrates their meromorphic continuation using dihedral GL(2) Maass forms, linking Fibonacci sequences with advanced modular form theory.

Contribution

It establishes a novel connection between Fibonacci-based Dirichlet series and dihedral GL(2) Maass forms, expanding the understanding of special functions in number theory.

Findings

Meromorphic continuation of generalized Fibonacci zeta functions

Connection between Fibonacci series and dihedral GL(2) Maass forms

Extension of classical zeta functions to new modular form contexts

Abstract

We show that a family of Dirichlet series generalizing the Fibonacci zeta function $\sum F(n)^{-s}$ has meromorphic continuation in terms of dihedral $\mathrm{GL}(2)$ Maass forms.