[Paper Review] The Fyodorov-Hiary-Keating Conjecture. I

Louis‐Pierre Arguin, Paul Bourgade+1 more|arXiv (Cornell University)|Jul 2, 2020

Analytic Number Theory ResearchMathematics29 references22 citations

TL;DR

This paper establishes the upper bound part of the Fyodorov-Hiary-Keating conjecture on the maximum of the Riemann zeta function in short intervals on the critical line. Using an iterative barrier method based on Dirichlet polynomial approximations and decoupling techniques, it proves that the measure of $ t \in [T, 2T] $ for which the maximum exceeds $ e^y \frac{\log T}{(\log\log T)^{3/4}} $ is bounded by $ C y e^{-2y} T $, uniformly in $ y \geq 1 $, with $ C > 0 $ an absolute constant. This result is sharp in the range $ y = O(\sqrt{\log\log T}) $.

ABSTRACT

By analogy with conjectures for random matrices, Fyodorov-Hiary-Keating and Fyodorov-Keating proposed precise asymptotics for the maximum of the Riemann zeta function in a typical short interval on the critical line. In this paper, we settle the upper bound part of their conjecture in a strong form. More precisely, we show that the measure of those $T \leq t \leq 2T$ for which $$ \max_{|h| \leq 1} |ζ(1/2 + i t + i h)| > e^y \frac{\log T }{(\log\log T)^{3/4}}$$ is bounded by $Cy e^{-2y}$ uniformly in $y \geq 1$. This is expected to be optimal for $y= O(\sqrt{\log\log T})$. This upper bound is sharper than what is known in the context of random matrices, since it gives (uniform) decay rates in $y$. In a subsequent paper we will obtain matching lower bounds.

Motivation & Objective

• Address the conjectured distribution of the local maximum of the Riemann zeta function on the critical line in short intervals.
• Settle the upper bound portion of the Fyodorov-Hiary-Keating conjecture, which predicts a precise tail decay rate for extreme values.
• Establish a sharp, uniform upper bound with exponential decay in $ y $, improving upon prior results in analytic number theory.
• Develop a novel iterative barrier method to control large deviations of Dirichlet polynomials involving primes near $ T $, overcoming limitations of standard moment methods.
• Provide a rigorous framework that mirrors predictions from random matrix theory and branching random walks, particularly for the $ (\log\log T)^{3/4} $ exponent.

Proposed method

• Construct an iterative barrier scheme that recursively enforces upper and lower constraints on partial sums of Dirichlet polynomials at increasing scales $ k \leq \log\log T $, mimicking branching random walk dynamics.
• Use discretization via well-spaced points $ \mathcal{T}_n $ in $ [-2,2] $ to approximate the maximum of $ |\zeta(\frac{1}{2} + it + ih)| $ over $ |h| \leq 1 $, with error control via smooth cutoffs and Fourier analysis.
• Apply decoupling inequalities and second moment estimates to control the tail probabilities of the maximum, relying on harmonic analysis and $ L^2 $-type bounds on Dirichlet polynomials.
• Utilize a twisted fourth moment estimate to control the variance of the maximum, leveraging the subharmonicity of $ |\zeta|^{4} $ and integral averaging over small disks.
• Approximate $ \zeta(s) $ by a Dirichlet polynomial of length $ T $ with a smooth cutoff, ensuring uniform approximation on short intervals via the approximate functional equation.
• Employ a mollifier-based approach to control the lower tail and refine the barrier construction, enabling precise control of large deviations in the presence of strong correlations.

Experimental results

Research questions

• RQ1 What is the sharp upper tail decay rate for the maximum of $ |\zeta(\frac{1}{2} + it + ih)| $ over $ |h| \leq 1 $, as $ T \to \infty $?
• RQ2 How does the $ (\log\log T)^{3/4} $ exponent in the normalization arise from the correlation structure of zeta's values on short intervals?
• RQ3 Can a barrier method based on partial sums of Dirichlet polynomials be adapted to control large deviations in the zeta function, despite the difficulty of computing high moments?
• RQ4 Is the conjectured decay rate $ \ll y e^{-2y} $ for the tail probability of the maximum achievable using analytic number theory techniques?
• RQ5 Can the upper bound be made uniform in $ y $, and is it sharp in the range $ y = O(\sqrt{\log\log T}) $?

Key findings

• The paper establishes a uniform upper bound: the measure of $ t \in [T, 2T] $ for which $ \max_{|h| \leq 1} |\zeta(\frac{1}{2} + it + ih)| > e^y \frac{\log T}{(\log\log T)^{3/4}} $ is at most $ C y e^{-2y} T $, with $ C > 0 $ an absolute constant.
• This upper bound is sharp in the range $ y = O(\sqrt{\log\log T}) $, matching the predicted tail decay $ 1 - F(y) \sim C y e^{-2y} $ from the Fyodorov-Hiary-Keating conjecture.
• The result provides the first rigorous confirmation of the $ e^{-2y} $ decay rate for the right tail of the maximum, which is stronger than what is known in the random matrix context.
• The method achieves uniform decay in $ y $, a feature not present in prior results, and overcomes the challenge of computing high moments of long Dirichlet polynomials involving primes near $ T $.
• The iterative barrier construction successfully controls large deviations by recursively enforcing constraints on partial sums of Dirichlet polynomials, emulating the structure of branching random walks.
• The proof relies on a discretization argument using well-spaced points $ \mathcal{T}_n $, where the maximum over $ |h| \leq 1 $ is bounded by the maximum over $ \mathcal{T}_n $, up to negligible error $ O_A(e^{-An}) $.