= Landau zero-free-region theorem
{c}
{title2=$1-\beta\gg\eta/(1+\log(M/\eta))$}
For $t\ge2$ and $0<\eta\le1/4$, suppose $|\zeta|\le M$, $M\ge2$, on radius-$\eta$ discs centred at $1+\eta/4+it$ and $1+\eta/4+2it$. Every zero $\beta+it$ has the displayed gap. The reciprocal <Euler product> bounds both centre values below by a constant times $\eta$. The <local logarithmic-derivative lemma> gives errors $B/\eta$, $B=1+\log(M/\eta)$. The <three-four-one zero-free-region argument> then implies $4/(\sigma-\beta)-3/(\sigma-1)\ll B/\eta$. For a zero close to one, choose $\sigma=1+6(1-\beta)$, making the left side $1/(14(1-\beta))$. Zeros farther away already satisfy the bound. Thus a logarithmic upper bound for zeta translates into a quantitative zero-free width.
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