New large value estimates for Dirichlet polynomials

TL;DR

This paper introduces a sharp large-values bound for Dirichlet polynomials of length $N$ at height $T$, addressing the critical regime near $V\approx N^{3/4}$ by combining a singular-value analysis of the matrix $M_W$ with a novel energy-based cancellation framework. The method blends non-stationary-phase cancellations with averaging over affine transformations and a detailed treatment of additive energy $E(W)$, yielding improved bounds for the frequency of large values and translating into a stronger zero-density bound for the zeta function. Consequently, it delivers new corollaries on primes in short intervals and almost-all short intervals, significantly advancing quantitative estimates tied to primes and the Riemann zeta function. The work also constructs a robust toolkit (S1,S2,S3 decomposition, affine-sum bounds, and energy controls) that can be applied to related large-value problems for Dirichlet polynomials.

Abstract

We prove new bounds for how often Dirichlet polynomials can take large values. This gives improved estimates for a Dirichlet polynomial of length $N$ taking values of size close to $N^{3/4}$, which is the critical situation for several estimates in analytic number theory connected to prime numbers and the Riemann zeta function. As a consequence, we deduce a zero density estimate $N(σ,T)\le T^{30(1-σ)/13+o(1)}$ and asymptotics for primes in short intervals of length $x^{17/30+o(1)}$.

Theorems & Definitions (72)

• Theorem 1.1: Large values estimate
• Theorem 1.2: Zero density estimate
• Corollary 1.3: Count of primes in short intervals
• Corollary 1.4: Count of primes in 'almost-all' short intervals
• Conjecture 1.5: Montgomery's large value conjecture
• Theorem 1.6: Heath-Brown, HB
• Lemma 1.7
• Proposition 3.1
• proof : Proof of Theorem \ref{['thrm:LargeValues']} assuming Proposition \ref{['prpstn:KeyProp']}
• Lemma 4.1: Large values of Dirichlet polynomials controlled by singular values