Montgomery’s large value conjecture for Dirichlet polynomials

Prove Montgomery’s large value conjecture: for any real parameter o > 1/2 and any Dirichlet polynomial D(t) = sum_{n~N} b_n e^{it log n} with coefficients satisfying |b_n| ≤ 1, if W ⊆ [0, T] is a 1-separated set such that |D(t)| > N^o for all t in W, then the cardinality satisfies |W| ≤ C(o) T^{o(1)} N^{2-2o}.

Background

The paper studies how often Dirichlet polynomials of length N can achieve large values on [0, T]. A classical lower bound shows that many examples have |W| ≳ N^{2-2o}, and Montgomery conjectured that this lower bound is essentially optimal. The authors present the conjecture explicitly as Conjecture 1.5 and discuss regimes where partial results are known, particularly for larger o, while noting that their new bounds improve key cases near o ≈ 3/4.

This conjecture directly influences quantitative estimates in analytic number theory, including zero-density bounds for the Riemann zeta function and distributions of primes. Establishing the conjecture would resolve the sharp frequency of large values across a broad range of parameters and unify several methods (mean value and large values) under a single optimal framework.