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The Fibonacci Zeta Function and Modular Forms

Eran Assaf, Chan Ieong Kuan, David Lowry-Duda, Alexander Walker

TL;DR

The paper establishes a meromorphic continuation for a family of Fibonacci-type zeta functions $Z_D^{\text{odd}}(s)$ and $Z_D^{\text{even}}(s)$ by expressing them as shifted convolution Dirichlet series tied to theta-coefficients and then performing a detailed spectral expansion on $\Gamma_0(4D)$ with nebentypus $\chi_{4D}$. A key step is regularizing growth at cusps via $V(z)=V_1(z)-E(z,\tfrac12)$, enabling a spectral decomposition of the Poincaré series into discrete and continuous parts; the continuous contribution vanishes and only dihedral Maass forms contribute to the discrete spectrum, after careful treatment of oldforms when $D\equiv1\pmod{4}$. The authors compute explicit inner products (Poincaré and dihedral) and Fourier coefficients, assemble these into a closed spectral expression, and verify that the resulting meromorphic continuation matches earlier Poisson-summation results. The even case is discussed via a regulated analogue $P_{-\ell}$, with the same spectral mechanism yielding a continuation in the left half-plane and aligning with established Poisson-transform expressions. Overall, the work connects Fibonacci-type zeta values to dihedral automorphic representations, providing explicit gamma-sum formulas and a robust modular-form framework for the continuations.

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.

The Fibonacci Zeta Function and Modular Forms

TL;DR

The paper establishes a meromorphic continuation for a family of Fibonacci-type zeta functions and by expressing them as shifted convolution Dirichlet series tied to theta-coefficients and then performing a detailed spectral expansion on with nebentypus . A key step is regularizing growth at cusps via , enabling a spectral decomposition of the Poincaré series into discrete and continuous parts; the continuous contribution vanishes and only dihedral Maass forms contribute to the discrete spectrum, after careful treatment of oldforms when . The authors compute explicit inner products (Poincaré and dihedral) and Fourier coefficients, assemble these into a closed spectral expression, and verify that the resulting meromorphic continuation matches earlier Poisson-summation results. The even case is discussed via a regulated analogue , with the same spectral mechanism yielding a continuation in the left half-plane and aligning with established Poisson-transform expressions. Overall, the work connects Fibonacci-type zeta values to dihedral automorphic representations, providing explicit gamma-sum formulas and a robust modular-form framework for the continuations.

Abstract

We show that a family of Dirichlet series generalizing the Fibonacci zeta function has meromorphic continuation in terms of dihedral Maass forms.

Paper Structure

This paper contains 15 sections, 7 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Shifted Convolution Dirichlet Series
  3. Spectral Expansion
  4. Subtracting growth at cusps
  5. Initial Spectral Decomposition
  6. Vanishing in the Continuous Spectrum
  7. Vanishing of Most of the Discrete Spectrum
  8. Orthogonalizing
  9. Explicitly computing remaining contributions
  10. Poincaré inner products
  11. Dihedral inner products with D = 2 mod 4
  12. Dihedral inner products with D = 1 mod 4
  13. Fourier coefficients
  14. Assembly
  15. Extended Remark on $Z_D^{\textup{even}}$

Key Result

Theorem 1

The $\mathcal{O}_D$ Fibonacci zeta functions admit meromorphic continuation to $s \in \mathbb{C}$. We have for all $s \in \mathbb{C}$ and for $\operatorname{Re} s < 0$, respectively.

Theorems & Definitions (13)

  • Theorem 1
  • Proposition 2: Proposition 6 of akldwFibonacciGeneral
  • Proposition 3
  • proof
  • Remark 4
  • Lemma 5
  • proof
  • proof : (Proof sketch when $\mu$ is a newform)
  • Lemma 6
  • proof
  • ...and 3 more