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 $$Z_F(s)=\sum _{n\ge 1}{F}_n^{-s},\mathrm{Re}s>0.$$ 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 $$Z_F({\sigma }_F)=4+2144^{-{\sigma }_F},$$ then $${\sigma }_F=0.743163398726901648\dots ,$$ $Z_F(s)\ne 0$ for $\mathrm{Re}s\ge {\sigma }_F$, while $$\overline{\{\mathrm{Re}\rho :Z_F(\rho )=0,\mathrm{Re}\rho >0\}}=[0,{\sigma }_F].$$ 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 $N\ge 12$ 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 $\log \phi /4$.

24 pages. Exact right edge of the zero set of the Fibonacci zeta function, with results on Jessen zero density, partial sums, Lucas zeta functions, and an entire q-Pochhammer completion. Reproducibility package: Zenodo, DOI 10.5281/zenodo.22300938