Majorant bound for the Dirichlet sum ∑_{n=N}^{2N} e^{it log n}

Determine whether the majorant Dirichlet polynomial D_major(t) = ∑_{n=N}^{2N} e^{it log n} satisfies |D_major(t)| ≤ T^{o(1)} N^{1/2} for all 1 ≤ |t| ≤ T.

Background

This majorant bound is a key input to Heath-Brown’s refined Hardy–Littlewood majorant principle adapted to difference sets; combined with that principle, it would imply sharp control on sums over differences and yield strong large-value estimates when the set of times has additive structure.

The conjectured bound reflects square-root cancellation in the exponential sum with logarithmic phase and is closely related to improvements in exponential sum estimates central to analytic number theory.