The Fibonacci Zeta Function and Continuation

2024/12/18 by Eran Assaf, Chan Ieong Kuan, Assaf, Eran +5
Mathematics · Physics and Astronomy · #11B39 #11M41 #30B40 #Advanced Mathematical Theories and Applications #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2412.13620

openalex publication_date 2024/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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