It seems to be that the simplicity of all the zeros is quite widely accepted as a working hypotheses, and it is known that a positive proportion are as such.
Titchmarsh explains in the last chapter on RH that the weak Mertens hypothesis, $M(x)=O(x^{1/2})$, implies that all zeros are simple. Though I think this is conjectured to be false by many authors. Is there a weaker bound whose optimality is known to imply that not all zeros are simple?
Given the classical estimate $N(\sigma, T)=O(T^{4\sigma(1-\sigma)+\epsilon})$ combined with the fact that the number of solutions to $\zeta(s)=z$ in any rectangle of height $T$ contained in $1/2\leq \sigma\leq 1$ is proportional to $T$, for any $z\neq 0$, simplicity of the zeros suggests that the derivative must vanish vastly more frequently in each such rectangle else one could presumably tie the number of such solutions to the number of zeros up to height $T$, for all sufficiently small $|z|$, by Rouche's theorem. One might therefore conjecture that $ \inf|\rho_{\zeta}-\rho_{\zeta'}|=0$ (is this known perhaps?).
An old theorem of MacDonald (Whittaker and Watson, p131) says that the number of zeros of an analytic function on the interior of a simple closed curve on which the modulus of $f$ is constant exceeds that of it's derivative by one. It can also be seen that the derivative necessarily vanishes on the largest such curve by the open and inverse mapping theorems. In the case of $\zeta(s)$ I'd expect that the largest of such curves around each zero would become small for large $T$, so I wonder if something in this direction is known?
Anyway, as Titchmarsh points out, $\Omega$-theorems for $M(x)$ seem to be more difficult to obtain than those for the prime counting functions and, since a significant distinction between the problems is that of the order of the zeros, I would like to enquire specifically about anything that is known in either direction, on any other hypotheses.
Best Answer
Quite a lot is known in the literature about zeros of $\zeta(s)$ and $\zeta^{\prime}(s)$, and in particular one can show that zeros of $\zeta^{\prime}(s)$ do get close to zeros of $\zeta(s)$. However, I am skeptical that these results have any implications for $M(x)$.
The number of zeros of $\zeta^{\prime}(s)$ with ordinates in $(0,T)$ is $$ = \frac{T}{2\pi} \log \frac{T}{4\pi e} + O(\log T). $$ This is due to Bruce Berndt, and is an application of the argument principle. Note that it is slightly different from the corresponding formula for $\zeta(s)$. The zeros of $\zeta^{\prime}$ are all expected to lie in the half plane $\sigma \ge 1/2$ -- this is equivalent to RH (proved first by Speiser). Levinson and Montgomery (Acta Math. 1974) investigated this further, and their work was the precursor to Levinson's famous result on the critical zeros of $\zeta(s)$. Levinson and Montgomery showed that most of the zeros of $\zeta^{\prime}(s)$ (with ordinate in $(0,T)$) lie in the region $|\sigma -1/2| \le w(T) \log \log T/\log T$ for any function $w(T)$ tending to infinity with $T$. In fact, most of the zeros of $\zeta^{\prime}$ probably lie closer to the critical line, on the scale of $1/\log T$ from $1/2$. For work in this direction, see papers of Conrey and Ghosh, Soundararajan, Yitang Zhang, Farmer and Ki, and most recently Radziwill (http://arxiv.org/abs/1301.3232, which will give the other references).
From above one knows that there are plenty of zeros of $\zeta^{\prime}(s)$ lying very close to the half line. From zero density results, we also know that there are plenty of zeros of $\zeta(s)$ lying close to the half line. For example, we can guarantee that there are at most $T/\log T$ zeros of $\zeta(s)$ with $|\sigma- 1/2| \ge 100 \log \log T/\log T$, say. The only thing is to rule out a conspiracy separating ordinates of such zeros of $\zeta$ and $\zeta^{\prime}$. A straightforward way to do this is to use Littlewood's theorem that the gaps between consecutive ordinates of zeros of $\zeta(s)$ tends to zero (at the rate of $1/\log \log \log |\gamma|$). First find, using the results above, $\gg T$ zeros of $\zeta^{\prime}$ (with ordinates in $[T,2T]$ say) such that these zeros all lie near the critical line, and their ordinates are spaced a distance $1$ apart. Next by Littlewood, for each such zero of $\zeta^{\prime}$, find a zero of zeta whose ordinate is close to it. Lastly, note that by the zero density theorem for zeta, at most $T/\log T$ of these corresponding zeros of zeta can be far from the $1/2$ line.
That completes the proof. The lazy argument above shows that $\min |\rho_{\zeta} -\rho_{\zeta^{\prime}}| \ll 1/\log \log \log T$ for zeros around ordinate $T$. By working harder one should be able to get $\ll 1/\log T$, and one probably cannot prove anything better. It is believed that zeros of $\zeta$ can get very close to each other, and so plausibly the truth here is that $|\rho_{\zeta} -\rho_{\zeta^{\prime}}|$ can get as small as $T^{-\frac 13+o(1)}$. This is very speculative, and even with this I don't see any implication for $M(x)$.