= Solution
Assume <Riemann hypothesis>. First fix $0<\delta<1/4$ and work on $\sigma_0=1/2+\delta$. We prove the <subpower zeta bound to the right of the critical line>, then move back to the line by the <functional equation> and <Phragmén–Lindelöf principle>.
Put $x=\log t$ for large positive $t$ and integrate the supplied smoothed <logarithmic derivative> identity horizontally from $\sigma_0$ to two. There are no zeros on this path under <Riemann hypothesis>, so the Euler-product logarithm at $2+it$ continues along it. Each prime-power term contributes at most $\Lambda(n)n^{-\sigma_0}/\log n\le n^{-\sigma_0}$, and the smoothing weights are at most one. Hence the integrated <prime> terms are bounded by
$$
C_\delta\sum_{n\le x^2}n^{-\sigma_0}\ll_\delta x^{1-2\delta}=o(\log t).
$$
All zeros have $\rho=1/2+i\gamma$. The local zero-count estimate from Question 3, with its reflected version for negative ordinates, gives uniformly for $\sigma_0\le\sigma\le2$
$$
\sum_\rho\frac1{|\rho-\sigma-it|^2}\le C_\delta\log t.
$$
To see the uniformity, sum the $O(\log(2+|\gamma|))$ zeros in successive unit ordinate intervals against $(\delta^2+|t-\gamma|^2)^{-1}$; the distant dyadic tails are summable. The zero-term numerator has modulus at most $x^{-2\delta}+x^{-\delta}$. Its integrated contribution is therefore at most $C_\delta x^{-\delta}\log t/\log x=o(\log t)$. The integrated supplied remainder is $O(x^{-1-\delta}\log t/\log x)$, also $o(\log t)$. Since $\log\zeta(2+it)$ is bounded, we obtain
$$
|\log\zeta(1/2+\delta+it)|=o_\delta(\log t).
$$
Thus for every fixed $\delta>0$ and $\eta>0$, $|\zeta(1/2+\delta+it)|\ll_{\delta,\eta}t^\eta$. Negative $t$ follow by <complex conjugation>.
The zeta <functional equation> and the gamma ratio give $|\zeta(1/2-\delta+it)|\ll_{\delta,\eta}t^{\delta+\eta}$. Zeta is <holomorphic> throughout this strip, since its <pole> at one is outside it, and Euler summation supplies <polynomial> vertical growth. The strip convexity conclusion of <Phragmén–Lindelöf principle> therefore gives at the midpoint
$$
|\zeta(1/2+it)|\ll_{\delta,\eta}(1+|t|)^{\delta/2+\eta}.
$$
For a prescribed $\varepsilon>0$, choose $\delta$ and $\eta$ with $\delta/2+\eta<\varepsilon$; the bounded $t$ range is harmless. This proves
$$
\boxed{\text{Riemann hypothesis}\ \Longrightarrow\ \text{Lindelöf hypothesis}.}
$$
The explicit-formula estimate is deliberately first made a fixed distance to the right of the <critical line>. No divergent zero bound at $\sigma=1/2$ is used.
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