Bound on off-diagonal entries of M_Dir M_Dir*

Establish that for the Dirichlet polynomial matrix M_Dir with entries (M_Dir)_{t,n} = n^{it}, every off-diagonal entry of the Gram matrix M_Dir M_Dir* has magnitude ≲ N^{1/2}.

Background

Sharp MM* control of off-diagonal entries would yield strong large-value bounds via the MM* method, especially in the regime σ > 3/4. For random matrices, such N^{1/2} bounds hold with high probability; for M_Dir this is conjectured but currently out of reach and tied to deep analytic number theory (Lindelöf).

Existing estimates (e.g., via van der Corput) give weaker T^{1/2}-type bounds, leading to partial large-value estimates but not the conjectured optimal N^{1/2} scale.