AI 中文总结
该研究确定了斐波那契ζ函数零点实部闭包的精确右边界，证明了相关零点性质，推导了部分和边界的渐近关系，还推广到卢卡斯ζ函数并构造了完备化整函数。
AI 中文摘要
设$F_1=F_2=1$，$F_{n+2}=F_{n+1}+F_n$，定义斐波那契ζ函数为$Z_F(s)=\sum_{n\ge1}F_n^{-s}$（$\operatorname{Re}s>0$）。我们确定了其绝对收敛半平面内零点实部闭包的精确右边界：若$\sigma_F$是方程$Z_F(\sigma_F)=4+2\cdot144^{-\sigma_F}$的唯一解，则$\sigma_F=0.743163398726901648\ldots$，当$\operatorname{Re}s\ge\sigma_F$时$Z_F(s)\neq0$，且正实部零点实部的闭包为$[0,\sigma_F]$。该边界在几乎周期意义上是尖锐的：在每条可允许竖线附近，零点的纵坐标相对密集。我们证明了边界附近存在增长维数的相位锁定和丢番图无零点尖点，描述了相关的Jessen函数和平滑平均垂直零点密度。对每个$N\ge12$的部分和，我们确定了对应精确闭包边界$\sigma_N$，证明$\sigma_N\nearrow\sigma_F$，并得到$\sigma_F-\sigma_N$的指数渐近式。我们还推导了正整数卢卡斯ζ函数的有限核心定理，以佩尔ζ函数作为显式例子。最后，利用已知的亚纯延拓，我们构造了一个自然的$q$-波赫哈默完备化，它是精确阶为2、型为$\log\varphi/4$的整函数。
英文摘要
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 $σ_F$ is the unique solution of $$ Z_F(σ_F)=4+2\,144^{-σ_F}, $$ then $$ σ_F=0.743163398726901648\ldots, $$ $Z_F(s)\neq0$ for $\operatorname{Re}s\geσ_F$, while $$ \overline{\{\operatorname{Re}ρ:Z_F(ρ)=0,\ \operatorname{Re}ρ>0\}}=[0,σ_F]. $$ 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 $N\ge12$ we determine the corresponding exact closure edge $σ_N$, prove $σ_N\nearrowσ_F$, 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 $\logφ/4$.
Comments24 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
