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The Fibonacci Zeta Function and Continuation

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

TL;DR

The paper constructs a family of Fibonacci-type zeta functions attached to real quadratic fields via traces of powers of the fundamental unit, unifying Fibonacci and Lucas sequences within the algebraic framework of $O_D$ and Pell-type parametrizations. Under $N(\\varepsilon)=-1$, it proves meromorphic continuation of the odd/even subsequence zeta functions through three distinct routes: (i) a direct binomial expansion, (ii) Poisson summation with appropriate regularization, and (iii) a modular-forms viewpoint via shifted-convolution Dirichlet series and Poincaré series. Each method reveals a precise pole structure and corroborates the other approaches, with the modular-form perspective linking the continuation to automorphic forms and spectral theory. Together, the results extend the Egami–Navas framework, connect algebraic number theory, analytic continuation, and modular forms, and set the stage for a deeper understanding of zeta functions arising from algebraic sequences in number fields.

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.

The Fibonacci Zeta Function and Continuation

TL;DR

The paper constructs a family of Fibonacci-type zeta functions attached to real quadratic fields via traces of powers of the fundamental unit, unifying Fibonacci and Lucas sequences within the algebraic framework of and Pell-type parametrizations. Under , it proves meromorphic continuation of the odd/even subsequence zeta functions through three distinct routes: (i) a direct binomial expansion, (ii) Poisson summation with appropriate regularization, and (iii) a modular-forms viewpoint via shifted-convolution Dirichlet series and Poincaré series. Each method reveals a precise pole structure and corroborates the other approaches, with the modular-form perspective linking the continuation to automorphic forms and spectral theory. Together, the results extend the Egami–Navas framework, connect algebraic number theory, analytic continuation, and modular forms, and set the stage for a deeper understanding of zeta functions arising from algebraic sequences in number fields.

Abstract

We introduce a family of Dirichlet series associated to real quadratic number fields that generalize the ordinary Fibonacci zeta function , where denotes the th Fibonacci number. We then give three different methods of meromorphic continuation to . Two are purely analytic and classical, while the third uses shifted convolutions and modular forms.

Paper Structure

This paper contains 8 sections, 9 theorems, 49 equations.

Table of Contents

  1. Introduction and Context
  2. Fibonacci and Lucas sequences from O(Q(sqrt(D)))
  3. Continuation via Binomial Expansion
  4. Continuation via Poisson Summation
  5. Odd-indexed Fibonacci zeta function
  6. Even-Indexed Fibonacci Zeta Function
  7. Continuation via Modular Forms
  8. A sketch of how to understand this continuation

Key Result

Theorem 1

The Fibonacci zeta function $Z_{\mathrm{Fib}}(s)$ has meromorphic continuation to all of $\mathbb{C}$, given explicitly by The function $Z_{\mathrm{Fib}}(s)$ has simple poles at $s = -2k + \frac{(2n + k)\pi i}{\log \varphi}$ for $k, n \in \mathbb{Z}$, $k \geq 0$, and is otherwise holomorphic.

Theorems & Definitions (19)

  • Theorem : Egami egami1999curious and Navas navas2001fibonacci, independently
  • Theorem 1
  • Remark 2
  • Example 3
  • Example 4
  • Example 5
  • Proposition 6
  • proof
  • Remark 7
  • Theorem 8
  • ...and 9 more