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PreprintCC BY 4.0arXiv

The Right Edge of the Zero Set of the Fibonacci Zeta Function

  • Marco Mantovanelli
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Abstract

Let F1=F2=1, Fn+2=Fn+1+Fn, and define the Fibonacci zeta function by ZF(s)=∑n≥1Fn−s, Res>0. We determine the exact right edge of the closure of the real parts of its zeros in the half-plane of absolute convergence. If σF is the unique solution of ZF(σF)=4+2 144−σF, then σF=0.743163398726901648…, ZF(s)≠0 for Res≥σF, while {Reρ:ZF(ρ)=0, Reρ>0}̅=[0,σF]. The edge is sharp in an almost-periodic sense: zeros occur with relatively dense ordinates near every admissible vertical line. We prove growing-dimensional phase locking near the edge and a Diophantine zero-free cusp, and describe the associated Jessen function and smooth mean vertical zero density. For every partial sum with N≥12 we determine the corresponding exact closure edge σN, prove σNnearrowσF, and obtain an exponential asymptotic for σF−σN. We also derive a finite-core theorem for positive integral Lucas zeta functions, with the Pell zeta function as an explicit example. Finally, using the known meromorphic continuation, we construct a natural q-Pochhammer completion that is entire of exact order $2$ and type logφ/4.

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Cite

BibTeXRIS
BibTeX
@misc{mantovanelli2026right,
  title = {{The Right Edge of the Zero Set of the Fibonacci Zeta Function}},
  author = {Marco Mantovanelli},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04993},
  eprint = {2609.04993},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04993},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Right Edge of the Zero Set of the Fibonacci Zeta Function
AU  - Marco Mantovanelli
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.04993
UR  - https://arxiv.org/abs/2609.04993
AB  - Let F_1=F_2=1, F_n+2=F_n+1+F_n, and define the Fibonacci zeta function by Z_F(s)=∑_n≥1F_n^−s, Res>0. We determine the exact right edge of the closure of the real parts of its zeros in the half-plane of absolute convergence. If σ_F is the unique solution of Z_F(σ_F)=4+2 144^−σ_F, then σ_F=0.743163398726901648…, Z_F(s)≠0 for Res≥σ_F, while {Reρ:Z_F(ρ)=0, Reρ>0}̅=[0,σ_F]. The edge is sharp in an almost-periodic sense: zeros occur with relatively dense ordinates near every admissible vertical line. We prove growing-dimensional phase locking near the edge and a Diophantine zero-free cusp, and describe the associated Jessen function and smooth mean vertical zero density. For every partial sum with N≥12 we determine the corresponding exact closure edge σ_N, prove σ_Nnearrowσ_F, and obtain an exponential asymptotic for σ_F−σ_N. We also derive a finite-core theorem for positive integral Lucas zeta functions, with the Pell zeta function as an explicit example. Finally, using the known meromorphic continuation, we construct a natural q-Pochhammer completion that is entire of exact order $2$ and type logφ/4.
ER  -