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

References

Conjecture [Montgomery's ℓq conjecture] Suppose N ≤ T ≤ N{O(1)}. For every q ≥ 2, ∥ M_{Dir} ∥_{∞ → q} ≲ N + N{1/2} T{1/q}.

In other words, for any q ≥ 2, if D(t) = ∑_{n=N}{2N} b_n e{i t log n} with |b_n| ≤ 1, then

∑_{t =1}T |D(t)|q ≲ Nq + N{q/2} T.

— Large value estimates in number theory, harmonic analysis, and computer science  (2503.07410 - Guth, 10 Mar 2025) in Section 3.4

In more classical language, these computations correspond to the fact that with current understanding of the large values of Dirichlet polynomials, we cannot rule out the scenario that

\left|\sum_{n \sim X{\beta_j}} \frac{a_j(n)}{n{1/2+it}}\right| \geq X{\beta_j(1/2-2/7)}\qquad j=1,2,3

for \gg X{2/7} one-separated points t\in[-X{\tau_1},X{\tau_1}], where a_1(n)=1 and |a_j(n)|\ll d_B(n) for j=2,3 and some fixed B.

In order to obtain the exponent 19/35-\varepsilon in Theorems \ref{thm:intro-mobius}, \ref{thm1.3}, \ref{thm1.4} and \ref{thm1.5}, we would need to essentially have the bound

\int_{X\varepsilon}{X{\tau_1+\varepsilon}}\left|\sum_{n \sim X{1/5}} \frac{1}{n{1/2+it}}\right|5 dt \ll \frac{X{1/2}}{\logA X},

and it is the above-mentioned scenario which prevents us from obtaining this.

— Higher order uniformity of the primes and cancellation of the Möbius function in shorter intervals  (2610.09567 - Matomäki et al., 7 Oct 2026) in Remark 4.3, Section 4 (A Heath-Brown–Iwaniec type trilinear estimate)