  In a recent paper, Gonek, Graham, and Lee introduced a notion of the
Lindel\"of hypothesis (LH) for general sequences which coincides with the usual
Lindel\"of hypothesis for the Riemann zeta function in the case of the sequence
of positive integers. They made two conjectures: that LH should hold for every
admissible sequence of positive integers, and that LH should hold for the
''generic'' admissible sequence of positive real numbers. In this paper, we
give counterexamples to the first conjecture, and show that the second
conjecture can be either true or false, depending on the meaning of
''generic'': we construct probabilistic processes producing sequences
satisfying LH with probability 1, and we construct Baire topological spaces of
sequences for which the subspace of sequences satisfying LH is meagre. We also
extend the main result of Gonek, Graham, and Lee, stating that the Riemann
hypothesis is equivalent to LH for the sequence of prime numbers, to the
context of Beurling generalized number systems.
