Montgomery’s ℓ^q conjecture for Dirichlet polynomials

Show that for the Dirichlet polynomial matrix M_Dir with entries (M_Dir)_{t,n} = n^{it} and any q ≥ 2, one has ∥M_Dir∥_{∞→q} ≲ N + N^{1/2} T^{1/q} uniformly for N ≤ T ≤ N^{O(1)}. Equivalently, for any Dirichlet polynomial D(t) = ∑_{n=N}^{2N} b_n e^{it log n} with |b_n| ≤ 1, the inequality ∑_{t=1}^T |D(t)|^q ≲ N^q + N^{q/2} T holds.

Background

This conjecture is a weaker but still powerful form of Montgomery’s large value conjecture. For each fixed N and T, there is a critical q where the two terms balance; proving the bound at this critical exponent implies the full range of q.

The ℓ^q bounds are closely connected to large-value estimates and zero-density results for the Riemann zeta function; proving them would yield the density hypothesis.