Rigorous first-hitting-time asymptotics for near-edge zeros

Establish rigorous upper bounds, or prove the predicted asymptotic behavior, for the least height at which the Fibonacci zeta function has a zero whose real part lies within a prescribed distance of the right edge; specifically, determine whether \(\log T_F(\varepsilon)\sim(4\sigma_F\log\varphi)^{-1}(\log(1/\varepsilon))^2\) as \(\varepsilon\downarrow0\).

Background

The paper proves a polynomial lower bound for the least height $T_F(\varepsilon)$ of a zero whose real part lies within $\varepsilon$ of the right edge $\sigma_F$. It also derives a shrinking-target model in which near-edge zeros correspond to a growing-dimensional Kronecker flow entering increasingly small phase neighborhoods.

The resulting heuristic predicts superpolynomial but subexponential growth, with $\log T_F(\varepsilon)$ asymptotic to $(4\sigma_F\log\varphi)^{-1}(\log(1/\varepsilon))^2$. The paper explains that proving this requires quantitative recurrence estimates for logarithms of primes in a dimension growing like $\log(1/\varepsilon)$, which are not supplied by the existing argument.