bibkey: mantovanelli2026fibonaccizeta authors: Marco Mantovanelli year: 2026 title: The Right Edge of the Zero Set of the Fibonacci Zeta Function doi: null url: https://arxiv.org/abs/2609.04993v1 claim: The preprint determines the right edge of the zero set of the Fibonacci zeta function and constructs an entire completion; the result concerns a Fibonacci Dirichlet series and does not transfer to the Riemann zeta function or Robin’s inequality. strata_touched: [] license: citation-only triage: anchor

The right edge of the Fibonacci zeta zero set

The source is arXiv:2609.04993v1, submitted 4 September 2026. The statements below are attributed to that version; its proof and numerical constants were not independently audited here, and no Lean verification is claimed.

It studies

$$Z_F(s)=\sum _{n\ge 1}{F}_n^{-s}$$

and determines the right edge of the closure of the real parts of its zeros in the half-plane of absolute convergence. If ${\sigma }_F$ is the unique solution of

$$Z_F({\sigma }_F)=4+2144^{-{\sigma }_F},$$

the paper reports

$${\sigma }_F=0.743163398726901648\dots ,$$

with no zeros of $Z_F$ for $\mathrm{Re}s\ge {\sigma }_F$ and zeros approaching every admissible vertical line to the left. It also gives finite-partial-sum edges, a Lucas-zeta core theorem, and an entire completion of order $2$.

Boundary of the RH interface

The Fibonacci growth in the exponents makes this source relevant to FIB recursive geometry, but $Z_F$ has not supplied the Euler product, von Mangoldt coefficients, or explicit formula for $\zeta (s)$. Its zero-free edge is therefore a theorem about a different almost-periodic Dirichlet series. No implication to the Riemann hypothesis, Robin’s $\sigma (n)/n$ bound, or the signed $\Phi$ tail follows from the edge value alone.

The usable interface is diagnostic: a FIB-generated zeta-like object can have a sharp zero boundary without sharing the arithmetic test set of $\zeta$. Any proposed FIB spectral proof of RH must exhibit an actual coefficient-preserving map to the Riemann zeta explicit formula.