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.

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