Littlewood established that $2e^{\gamma} \geq \limsup_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be generally believed that the lower bound represents the truth, and even, in the most optimistic form, that quite possibly there is an absolute constant $C$ such that $|\zeta(1+it)| \leq e^{\gamma}(\log{\log{t}} + \log{\log{\log{t}}}+C)$ for all $t > 10$.

Suppose in these questions that we replace $|\zeta(1+it)|$ by $$ B(t) := \sup_k \Big| \sum_{n=1}^k n^{-1-it} \Big|, \quad \textrm{ resp. } \widetilde{B}(t):= \sup_{m,k} \Big| \sum_{n=m}^k n^{-1-it} \Big|, $$ the largest partial sums, respectively the largest subsum over all intervals.

Does Littlewood's RH result extend to $B$ or even $\widetilde{B}$? That is: should the exponential sums bound $\widetilde{B}(t) = O(\log{\log{t}})$ be possible to prove on RH, or what kind of bound is available in this uniformity? Should it be reasonable to expect the strongest possible bound $\widetilde{B}(t) \leq e^{\gamma}(\log{\log{t}} + \log{\log{\log{t}}}+C)$, for all $t$?

There are evident versions of this question for $L(1,\chi)$, $\frac{\zeta'}{\zeta}(1+it)$ and $\frac{L'}{L}(1,\chi)$. In the last of these, the strongest bound seems to tie well with the belief that the smallest quadratic non-residue mod $q$ is $\ll \log{q}\log{\log{q}}$ -- which it certainly implies, at least when $q$ is prime.