I think it is probably well known that, for every $1/2<\sigma\leq 1$, the function $1/\zeta(\sigma+it)$ is unbounded.
Yet, I cannot decide how deep this is. I imagine it could be proved using a universality argument, or by extending the $\sigma=1$ case by a convexity argument on the $\mu$-function, or perhaps directly by Kronecker's theorem, or maybe even Euler-MacLaurin, but I don't yet know.
Therefore I would like to know about proofs and, importantly, anything you would like to say about how "deep" this is. For example, can you explain that it follows from something quite trivial, or perhaps that it is relatively as weak/strong as something else?