= Local logarithmic-derivative lemma
{title2=$f'/f=\sum_{|\rho-z_0|\le R/2}(z-\rho)^{-1}+O((1+\log(M/|f(z_0)|))/R)$}
If $f$ is <holomorphic> near the closed radius-$R$ disc, $f(z_0)\ne0$, and $|f|\le M$, the displayed estimate holds away from zeros for $|z-z_0|\le R/3$, counting zeros with multiplicity. Factoring local zeros isolates their poles; the remaining logarithm is controlled by disc estimates. If all zeros lie to the left of a point $z$ in real part, their real contributions are nonnegative. This is the disc estimate used by the <Landau zero-free-region theorem>; a version with general radius ratios is Lemma 24.17 in https://personal.science.psu.edu/rcv4/Vol3/Vol3.pdf[Montgomery and Vaughan].
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