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

Abstract

Let $F_1=F_2=1$, $F_{n+2}=F_{n+1}+F_n$, and define the Fibonacci zeta function by $$ Z_F(s)=\sum_{n\ge1}F_n^{-s},\qquad \operatorname{Re}s>0. $$ We determine the exact right edge of the closure of the real parts of its zeros in the half-plane of absolute convergence. If $\sigma_F$ is the unique solution of $$ Z_F(\sigma_F)=4+2\,144^{-\sigma_F}, $$ then $$ \sigma_F=0.743163398726901648\ldots, $$ $Z_F(s)\neq0$ for $\operatorname{Re}s\ge\sigma_F$, while $$ \overline{\{\operatorname{Re}\rho:Z_F(\rho)=0,\ \operatorname{Re}\rho>0\}}=[0,\sigma_F]. $$ 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 $N\ge12$ we determine the corresponding exact closure edge $\sigma_N$, prove $\sigma_N\nearrow\sigma_F$, and obtain an exponential asymptotic for $\sigma_F-\sigma_N$. 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 $\log\varphi/4$.