[Paper Review] New large value estimates for Dirichlet polynomials

Larry Guth, James Maynard|arXiv (Cornell University)|May 31, 2024

Analytic Number Theory Research4 citations

TL;DR

This paper establishes new large value estimates for Dirichlet polynomials, improving bounds for when such polynomials take values near $ N^{3/4} $, which is critical in analytic number theory. The key result is a refined bound on the number of large values, leading to improved zero density estimates for the Riemann zeta function and stronger asymptotic formulas for primes in short intervals, including exponents $ 17/30 $ and $ 2/15 $, surpassing prior results by Huxley.

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 $x^{17/30+o(1)}$.

Motivation & Objective

• To improve the known bounds on how often Dirichlet polynomials of length $ N $ can take large values, particularly near the critical threshold $ V hickapprox N^{3/4} $.
• To refine zero density estimates for the Riemann zeta function by leveraging improved large value bounds for Dirichlet polynomials.
• To strengthen asymptotic estimates for the number of primes in short intervals, improving the range of validity beyond previous results.
• To provide a quantitative advancement in analytic number theory by addressing the limiting case where classical methods fail to give optimal bounds.

Proposed method

• The authors derive a new large values estimate for Dirichlet polynomials with coefficients bounded by 1, using a refined analysis of the distribution of large values over $ T $-intervals.
• They combine this with the classical Mean Value Theorem and the Montgomery-Halasz-Huxley large values estimate, but improve the trade-off in the critical range $ N^{7/10+ heta} < V < N^{8/10- heta} $.
• The main bound is expressed as $ R \igl{(}N^{2}V^{-2} + N^{18/5}V^{-4} + TN^{12/5}V^{-4}\bigr{)} $, where $ R $ is the number of $ 1 $-separated points where the polynomial exceeds $ V $.
• The method relies on a novel application of $ L^p $-type estimates and orthogonality arguments, particularly in the regime where $ V \approx N^{3/4} $.
• The improved large value bound is then applied to the explicit formula for the von Mangoldt function to control error terms in prime counting functions.
• By choosing appropriate $ T $-dependent parameters and combining with zero-free regions and Vinogradov-Korobov bounds, the authors derive improved error terms in short interval prime counts.

Experimental results

Research questions

• RQ1 What is the optimal upper bound on the number of $ 1 $-separated points in $[0,T]$ where a Dirichlet polynomial of length $ N $ takes values $ \geq V $, especially when $ V \approx N^{3/4} $?
• RQ2 How can improved large value estimates for Dirichlet polynomials be used to strengthen zero density estimates for the Riemann zeta function?
• RQ3 What is the best possible exponent $ \alpha $ such that $ \pi(x+y) - \pi(x) \sim y / \log x $ holds for $ y = x^{\alpha + o(1)} $?
• RQ4 Can the critical case $ V \approx N^{3/4} $, which limits previous results, be handled with a new method yielding better quantitative bounds?

Key findings

• The paper establishes a new large values estimate: $ R \leq T^{o(1)}\bigl{(}N^{2}V^{-2} + N^{18/5}V^{-4} + TN^{12/5}V^{-4}\bigr{)} $, which improves on classical bounds in the range $ N^{7/10+\epsilon} < V < N^{8/10-\epsilon} $ and $ N \leq T^{5/6-\epsilon} $.
• The improved large value bound leads to a zero density estimate $ N(\sigma,T) \ll T^{30(1-\sigma)/13 + o(1)} $, improving on Huxley’s exponent $ 12/5 $.
• For primes in short intervals, the paper shows that $ \pi(x+y) - \pi(x) = \frac{y}{\log x} + O\bigl{(}y \exp(-\sqrt[4]{\log x})\bigr{)} $ holds for $ y \in [x^{17/30+\epsilon}, x^{0.99}] $, improving Huxley’s $ 7/12 $.
• For 'almost-all' short intervals, the result holds for $ y \in [X^{2/15+\epsilon}, X^{0.99}] $, improving Huxley’s $ 1/6 $.
• The improvement arises from a sharper control of the contribution of Dirichlet polynomials of size $ \approx N^{3/4} $, which was the bottleneck in earlier methods.
• The method applies to a broad class of problems in analytic number theory where Dirichlet polynomials and their large values play a central role.