Montgomery’s large value conjecture for Dirichlet polynomials

Establish that for the Dirichlet polynomial matrix M_Dir with entries (M_Dir)_{t,n} = n^{it}, if the coefficients satisfy |b_n| ≤ 1 and σ > 1/2, then for all N and T with N ≤ T ≤ N^{O(1)} the large-value functional satisfies LV_{M_Dir, ℓ^∞}(N^σ) ≲ N^{2 − 2σ}. Equivalently, for any Dirichlet polynomial D(t) = ∑_{n=N}^{2N} b_n e^{it log n} with |b_n| ≤ 1 and any set W ⊂ Z ∩ [0,T] on which |D(t)| > N^σ, one must have |W| ≲_σ N^{2 − 2σ}.

Background

The conjecture refines an earlier (now false) conjecture by Montgomery by imposing the ℓ^∞ normalization |b_n| ≤ 1, which rules out counterexamples based on approximate geometric series. It predicts that Dirichlet polynomials behave, in large-value regimes, similarly to random matrices in terms of the number of large outputs.

If true, this conjecture implies strong density bounds on zeros of the Riemann zeta function (the ‘density hypothesis’), giving near–Riemann–hypothesis strength bounds for prime distribution in short intervals.