= Solution
Put $L=\log t$ and $\ell=\log L$, for sufficiently large $t$. Apply the <Landau zero-free-region theorem> with
$$
\eta=a(\ell/L)^{2/3},
$$
where $a>0$ is fixed and small enough that $\eta\le1/4$. On its two discs, the real part is at least $1-3\eta/4$ and the imaginary part is comparable to $t$. The given <Richert bound for the Riemann zeta function> therefore gives, on the part left of one,
$$
\log|\zeta(z)|\le C'\eta^{3/2}L+\tfrac23\log L+O(1)=O(\ell).
$$
On the part right of one, the separately given $O(\log^{2/3}t)$ bound gives the same conclusion. We may thus choose $M=L^A$ for one fixed sufficiently large $A$. Also $\log(1/\eta)=O(\ell)$, so the logarithmic term in the <Landau zero-free-region theorem> is $O(\ell)$. Its conclusion is
$$
\boxed{\zeta(\sigma+it)\ne0\quad\text{for}\quad\sigma\ge1-\frac c{(\log t)^{2/3}(\log\log t)^{1/3}}}
$$
for large $t$ and a sufficiently small positive $c$. <Complex conjugation> supplies negative heights. This is the <Vinogradov-Korobov zero-free region>. Only the stated Richert upper bounds, the <Euler product>, the <pole> at one and the proved <Landau zero-free-region theorem> were used; no prior zero-free-region theorem was assumed.
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