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The Fyodorov--Hiary--Keating Conjecture on Mesoscopic Intervals

This paper establishes precise upper bounds for the maximum and second moment of the Riemann zeta function on mesoscopic intervals, thereby settling the upper bound of the Fyodorov–Hiary–Keating conjecture and generalizing previous results by Harper through an adaptation of a recursive scheme.

Original authors: Louis-Pierre Arguin, Jad Hamdan

Published 2026-05-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Louis-Pierre Arguin, Jad Hamdan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Riemann zeta function, $\zeta(s)$ , as a vast, chaotic, and infinitely complex landscape. Mathematicians have long been fascinated by the "heights" of this landscape, specifically along a central line called the "critical line." The question is: How high can the peaks get in a small, specific neighborhood?

This paper by Louis-Pierre Arguin and Jad Hamdan is like a team of expert surveyors who have finally built a precise ruler to measure the tallest peaks in these neighborhoods. They are testing a famous prediction (the Fyodorov–Hiary–Keating conjecture) about exactly how high these peaks can climb and how often they reach those heights.

Here is a breakdown of their work using simple analogies:

1. The Landscape and the "Short Interval"

Think of the zeta function as a mountain range that stretches forever.

The Critical Line: This is the main ridge we are walking along.
The Short Interval: Instead of looking at the whole mountain range, the authors look at a tiny, specific patch of the ridge. They call this a "mesoscopic interval." It's not just a single point, but a small stretch of the path.
The Goal: They want to know the maximum height ( $\max$ ) of the zeta function within this tiny stretch.

2. The "Branching Random Walk" (The Forest Analogy)

To understand the zeta function, the authors use a model called a Branching Random Walk. Imagine a tree growing in a forest:

The Roots: The base of the tree represents the start of the interval.
The Branches: As the tree grows, it splits into many branches. Each branch represents a different point on the interval.
The Height: The height of a leaf at the top of a branch represents the value of the zeta function at that point.
The Connection: Because the branches split from the same trunk, they are "correlated." If one branch is high, its neighbors are likely high too. This is similar to how the zeta function behaves: nearby points tend to have similar values.

The authors realized that in these short intervals, the zeta function behaves exactly like this tree. The "trunk" of the tree (the part shared by all branches) is a specific mathematical term they call $P_\tau(\theta)$ . The "leaves" (the unique parts of the branches) behave like a random walk.

3. The Main Discovery: The "Upper Bound"

The authors proved a strict limit on how high the zeta function can go in these intervals.

The Prediction: The Fyodorov–Hiary–Keating conjecture predicted that the height of the peak follows a specific formula involving a "Gaussian" (a bell curve) and a "Gumbel" distribution (a curve that describes extreme events, like the highest flood level in a century).
The Result: The authors proved that the probability of the zeta function exceeding a certain height is extremely low. They showed that the height is bounded by a formula that looks like:
$\text{Height} \approx \text{Average} + \text{Random Fluctuation} + \text{Extreme Tail}$
They confirmed that the "Random Fluctuation" part is indeed a standard bell curve (Gaussian), and they calculated the exact probability of the "Extreme Tail" (the super-high peaks).

In simple terms: They proved that while the zeta function can get very high, it almost never exceeds a specific "ceiling" that they calculated. If it does, it's a one-in-a-million (or much rarer) event.

4. The "Partition Function" (The Energy of the System)

The paper also looks at the second moment of the zeta function.

The Analogy: Imagine the zeta function as a system of energy. The "partition function" is like the total energy of the system.
The Freezing Transition: The authors studied what happens when you look at the "energy" of the system. They found that at a certain "critical temperature" (mathematically, a specific parameter $\beta=2$ ), the system undergoes a "freezing transition."
The Result: They proved an upper bound for this energy. They showed that the energy is dominated by two factors:
1. A "log-normal" factor (related to the shared trunk of the tree).
2. A "square root" correction factor (related to the unique branches).
  This confirms a previous guess by Fyodorov and Keating that these two factors multiply together to determine the total energy.

5. How They Did It: The "Recursive Barrier"

How did they manage to measure these peaks so precisely?

The Problem: The zeta function is too complex to measure directly. It's like trying to measure the height of every leaf on a massive tree without climbing it.
The Solution: They used a "recursive scheme" (a step-by-step method) involving barriers.
- Imagine placing invisible fences (barriers) at different heights as you move up the tree.
- They proved that if the tree stays below these fences at every step, the final height cannot be too high.
- They used a "partial barrier" strategy. Instead of checking the whole tree at once, they checked it in stages. They placed a strong fence in the first half of the tree's growth and a looser one in the second half. This allowed them to control the "error" and prove the limit with high precision.

Summary

This paper is a major step forward in understanding the "extreme weather" of the Riemann zeta function.

What they did: They built a precise mathematical ruler to measure the tallest peaks of the zeta function in short intervals.
What they found: They confirmed that the peaks follow a specific pattern predicted by a famous conjecture. The peaks are a mix of a standard random fluctuation and a rare, extreme event.
Why it matters: It settles a long-standing debate about the "upper bound" of these peaks, proving that the zeta function behaves in a very specific, predictable way even in its most chaotic moments.

They didn't just guess the height; they proved that the zeta function cannot climb higher than their calculated ceiling, and they explained exactly why it behaves that way using the "tree" (branching random walk) analogy.

Technical Summary: The Fyodorov–Hiary–Keating Conjecture on Mesoscopic Intervals

Problem Statement
This paper addresses the distribution of the maximum and the second moment (partition function) of the Riemann zeta function, $\zeta(s)$ , on the critical line $\Re(s) = 1/2$ over "mesoscopic" short intervals. Specifically, the authors investigate the behavior of $\max_{|h| \le (\log T)^\theta} |\zeta(1/2 + i(\tau + h))|$ and the integral of its square, where $\tau$ is uniformly distributed in $[T, 2T]$ and $\theta \in (-1, 0]$ .

The central problem is to verify the Fyodorov–Hiary–Keating (FHK) conjecture, which predicts that the local maxima of $\log |\zeta|$ on such intervals behave like the maximum of a branching random walk (BRW). While the leading order behavior was previously established for $\theta < 0$ (by Arguin, Ouimet, and Radziwiłł) and the full distribution for $\theta = 0$ (by Arguin, Bourgade, and Radziwiłł), the case for $\theta \in (-1, 0)$ remained open regarding the precise sub-leading order and the explicit identification of the Gaussian fluctuation term.

Methodology
The authors employ a probabilistic approach rooted in the connection between the zeta function and branching random walks, utilizing a recursive scheme adapted from Arguin, Bourgade, and Radziwiłł [6]. The methodology involves several key technical components:

Decomposition of the Zeta Function: The authors decompose $\log |\zeta(1/2 + i(\tau + h))|$ into a "long" Dirichlet polynomial part, $S_{t_\theta}(h)$ , which captures correlations up to primes $p \le \exp((\log T)^{|\theta|})$ , and a "short" shifted part, $S_k(h) = S_{t_\theta+k}(h) - S_{t_\theta}(h)$ .
- For $\theta < 0$ , the term $S_{t_\theta}(0)$ acts as a common factor (roughly constant over the interval) that introduces a Gaussian fluctuation.
- The remaining term is modeled as a BRW on a tree of depth $t(1+\theta)$ .
Recursive Barrier Scheme: To bound the maximum, the authors construct a sequence of "good sets" $G_\ell$ defined by upper and lower barriers ( $U_y(k)$ and $L_y(k)$ ) that the partial sums $S_k(h)$ must satisfy at specific checkpoints $t_\ell$ . This scheme, originally developed for BRWs by Bramson [15], allows for the control of large deviations by restricting the range of Dirichlet polynomials to lengths where moment estimates (via the mean-value theorem) remain valid.
Partial Barrier Strategy for the Partition Function: For the second moment (partition function) at criticality ( $\beta=2$ ), the authors introduce a "partial barrier" strategy. Unlike previous approaches that required barriers over the full time interval, they demonstrate that restricting the process up to time $t^* = (t - t_\theta)/2$ is sufficient to recover the correct normalization factor. This avoids the technical difficulties of controlling long Dirichlet polynomials over the entire interval.
Correlation Control: A critical technical lemma (Proposition 1.5) establishes that the contribution from primes $p \le \exp((\log T)^{|\theta|})$ is essentially constant over the interval $|h| \le (\log T)^\theta$ , with deviations decaying as a Gaussian tail. This justifies factoring out the term $|P_\tau(\theta)|$ (related to $e^{-S_{t_\theta}(0)}$ ) from the maximum and moment estimates.

Key Contributions and Results

Upper Bound for the Local Maximum (Theorem 1.2):
The authors prove a sharp upper bound for the maximum of $\zeta$ on mesoscopic intervals. They show that for fixed $\theta \in (-1, 0]$ and large $T$ , the probability that the maximum exceeds a threshold $y$ is bounded by:
$P\left( \max_{|h| \le (\log T)^\theta} |\zeta(1/2 + i(\tau + h))| > e^y \frac{(\log T)^{1+\theta}}{(\log \log T)^{3/4}} \right) \le C y \exp\left(-2y - \frac{y^2}{(1+\theta)\log \log T}\right)$
Crucially, this result explicitly identifies the source of the Gaussian term in the FHK conjecture as the quantity $S(t) = S_{t_\theta}(0)$ , which converges in distribution to a standard Gaussian. This settles the upper bound of the conjecture up to a multiplicative constant for $\theta \in (-1, 0]$ .
Upper Bound for the Partition Function (Theorem 1.3):
The paper derives an upper bound for the second moment $Z_2(\theta)$ on mesoscopic intervals. The result shows that the probability of $Z_2(\theta)$ exceeding a threshold decays as $A^{-1+\epsilon}$ . The bound features a product of a log-normal factor (arising from the long primes) and a square-root correction term $\sqrt{\log \log T}$ (arising from the short primes), generalizing Harper's result for $\theta=0$ .
$P\left( Z_2(\theta) |P_\tau(\theta)|^2 \ge A \frac{(\log T)^{1+\theta}}{\sqrt{(1+\theta)\log \log T}} \right) \le C A^{-1+\epsilon}$
Constrained Large Deviation Estimates (Theorem 1.4):
The authors establish large deviation estimates for $\log |\zeta|$ constrained by barriers on a prime interval. This result, which generalizes techniques from [1] and [6], is of independent interest and serves as the technical engine for the main theorems. It relates the probability of the zeta function exceeding a value $V$ under barrier constraints to the corresponding probability for a Gaussian random walk.

Significance
The paper claims to settle the upper bound of the Fyodorov–Hiary–Keating conjecture for mesoscopic intervals ( $\theta \in (-1, 0]$ ) with $O(1)$ precision, a significant improvement over previous leading-order results. By explicitly identifying the Gaussian fluctuation term $S(t)$ , the work clarifies the structure of the maximum, confirming that it is the sum of a Gaussian variable (from long primes) and the maximum of a branching random walk (from short primes).

Furthermore, the paper provides a new proof of Harper's bound for the critical second moment at $\theta = 0$ and generalizes it to $\theta < 0$ . The authors note that their proof differs from Harper's approach by utilizing a "partial barrier" strategy, which allows for rigorous control of the measure of high points without relying on the full complexity of the barrier over the entire interval. The work bridges the gap between the known results for $\theta=0$ (random matrix theory analogues) and $\theta < 0$ , providing a unified framework for understanding the extreme values of the zeta function across different scales.