# A zero-free strip wide enough for rung 227 The exact-rational checker [`certify_227_strip.py`](certify_227_strip.py) proves a slightly stronger analytic zero-free strip for the completed Davenport–Heilbronn function. It uses the exact functional equation `F(s)=F(1−s)` and the period-five coefficients `(1, κ, −κ, −1, 0)`. The positive root of `κ²+2φκ−1=0`, `φ=(1+√5)/2`, obeys `κ<2/7`: indeed `√5>31/14` gives `φ>45/28`, so that polynomial is positive at `κ=2/7`. Set `σ=2799/2000=1.3995`. For `Re(s)≥σ`, the absolute Dirichlet series after its leading `1` is bounded by ``` Σ(n=2..50) c_n n^(−σ) + (18/7) [50^(−σ) + 50^(1−σ)/(5(σ−1))], ``` where `c_n=(1,2/7,2/7,1,0)` periodically. Each complete block of the tail, starting at `n=51`, has coefficient mass `18/7`; replacing its five denominators by `(5m)^σ` and applying the first-term-plus-integral bound yields the displayed tail. The checker encloses each irrational power above by `10^12/m`, where the two exact integer comparisons `m^2000 ≤ n^p·10^(12·2000) < (m+1)^2000` are verified. Its floating root estimate only accelerates the search; all decisive inequalities use integers and rational fractions. The resulting upper bound is ``` 0.998837088389815590325288... < 999/1000 < 1. ``` Therefore `f(s)≠0` for `Re(s)≥2799/2000`; completion has no prefactor zeros or poles there. Reflection gives the completed-zero strip `−799/2000 < Re(ρ) < 2799/2000`, hence `|Re(ρ)−1/2|<1799/2000`. For `q=ρ(1−ρ)` and `Im(ρ)=t≥260`, the ratio between the absolute imaginary and positive real parts of `q` is less than ``` 2(1799/2000)t / [t²+1/4−(1799/2000)²] ≤ 1870960000/270397763599. ``` The middle expression decreases for `t≥260`. For real `x>0`, `|arg(4xq/(x+q)²)|≤|arg q|