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Brownian behaviour of the Riemann zeta function around the critical line

This paper establishes a Brownian extension to Selberg's central limit theorem for the Riemann zeta function, demonstrating that the distribution of its maximum value along the critical line converges to a limiting form analogous to the reflection principle for Brownian motion.

Original authors: Louis Vassaux

Published 2026-06-02

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CC BY 4.0

Original authors: Louis Vassaux

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, $\zeta(s)$ , not as a static equation on a chalkboard, but as a wild, unpredictable river flowing through a complex landscape. For over a century, mathematicians have been trying to understand the "turbulence" of this river, especially near a mysterious boundary known as the critical line.

Louis Vassaux's paper is like a new set of goggles that allows us to see that this river doesn't just flow randomly; it behaves exactly like a specific type of random motion known as Brownian motion (or a "random walk").

Here is the breakdown of the paper's discoveries using everyday analogies:

1. The Big Discovery: The River is a Random Walk

For a long time, we knew that if you pick a random spot on the critical line, the value of the zeta function looks like a "Gaussian" (a bell curve). This was a major result by Selberg.

Vassaux takes this a step further. He looks at what happens if you move horizontally away from the critical line, step by step. He proves that as you zoom out to look at the big picture, the path the zeta function takes looks exactly like a drunkard's walk (Brownian motion).

The Analogy: Imagine a drunk person stumbling along a straight line. If you watch them for a long time, their path is jagged and unpredictable. Vassaux shows that the zeta function, when viewed from a specific angle and scale, stumbles in the exact same mathematical way as that drunk person. It's not just a random number; it's a random path.

2. The "Reflection Principle": Bouncing Off the Ceiling

One of the most famous rules about Brownian motion is the Reflection Principle. It says that if a random walker starts at zero, the probability of them reaching a high peak is related to the probability of them ending up at a specific point.

Vassaux shows that the zeta function obeys this same rule.

The Analogy: Imagine a ball bouncing inside a tube. If you ask, "What is the chance this ball hits the ceiling?" the answer depends on the same math that tells you "Where is the ball right now?"
The Result: The paper proves that the maximum height the zeta function reaches as you move away from the critical line follows a specific distribution (the absolute value of a Gaussian). It's like saying, "We can predict the highest wave the river will make, even if we can't predict the exact shape of the water at any single second."

3. The "Arcsine Law": Where Does the River Spend Its Time?

The paper also looks at how much time the zeta function spends "above" a certain level versus "below" it as you move along the horizontal path.

The Analogy: Imagine a pendulum swinging back and forth. You might expect it to spend half its time on the left and half on the right. But for a random walk, it's actually more likely to spend most of its time on one side or the other, rather than switching back and forth constantly. This is called the Arcsine Law.
The Result: Vassaux shows that the zeta function behaves this way too. It tends to "stick" to one side of the line for long stretches before flipping, rather than oscillating perfectly evenly.

4. The "Law of the Iterated Logarithm": The Limits of the Stumble

Finally, the paper looks at the extreme limits of how far the zeta function can wander from its starting point.

The Analogy: If a drunk person walks for a very long time, there is a mathematical limit to how far they can stray from the starting point relative to the time they've been walking. They can't wander off to infinity instantly; their wandering speed is constrained by a specific formula involving "logarithms" (a way of measuring growth).
The Result: The paper proves that the zeta function respects this same limit. It confirms that the "stumbles" of the zeta function are wild, but they are wild in a very controlled, predictable mathematical way.

How Did They Prove It?

To prove all this, the authors had to do some heavy lifting:

Simplifying the Monster: The zeta function is incredibly complex. They replaced it with a simpler "Dirichlet sum" (a type of infinite series) that acts like a shadow of the real thing but is easier to calculate.
Checking the Steps: They proved that this simplified version behaves like a random walk.
Handling the "Zeros": The zeta function has "zeros" (points where it equals zero) that act like potholes in the road. The authors had to prove that these potholes are rare enough that they don't ruin the overall "drunk walk" pattern. They showed that if you avoid a tiny, specific set of bad spots, the rest of the path is perfectly smooth and random.

Summary

In short, this paper establishes that the Riemann zeta function, when viewed horizontally near its critical line, is not just a chaotic mess. It is a Brownian motion.

This is a powerful realization because it means we can use the entire, well-understood toolbox of probability theory (which describes how particles move in a fluid, how stock prices fluctuate, or how a drunk person walks) to understand the deepest mysteries of prime numbers and the zeta function. We now know that the "chaos" of the zeta function follows the same rules as the randomness of nature itself.

Technical Summary: Brownian Behaviour of the Riemann Zeta Function Around the Critical Line

Problem Statement
The paper addresses the statistical behavior of the Riemann zeta function, $\zeta(s)$ , in the vicinity of the critical line $\text{Re}(s) = 1/2$ . While Selberg's Central Limit Theorem (CLT) establishes that $\log \zeta(1/2 + i\tau)$ behaves like a complex Gaussian random variable for $\tau$ chosen uniformly from $[0, T]$ , the behavior of $\zeta(s)$ when moving horizontally away from the critical line (i.e., varying the real part $\sigma$ ) remains less understood. Specifically, the author investigates the functional convergence of the process defined by moving horizontally from the critical line into the critical strip as $T \to \infty$ . The central question is whether this process converges to a Brownian motion, thereby allowing the application of probabilistic tools (such as the reflection principle and the law of the iterated logarithm) to the zeta function.

Methodology
The proof of the main theorem relies on establishing two conditions for the convergence in law of the stochastic process $Z(T)$ :

Convergence of Finite-Dimensional Distributions: The author demonstrates that the joint distribution of $\log \zeta$ at multiple points along a horizontal path converges to a multivariate complex Gaussian vector. This is achieved by approximating $\log \zeta$ with a truncated Dirichlet series and applying Lemma 2.2 (adapted from [4]), which provides conditions under which sums of prime powers converge to Gaussian variables. The covariance structure is derived using properties of prime sums (Lemma 3.3).
Tightness: To upgrade finite-dimensional convergence to functional convergence in the space of continuous functions $C_0([0, 1], \mathbb{C})$ $C_{0} ([0, 1], C)$ , the author employs a modified Kolmogorov tightness criterion (Theorem 2.3). This requires bounding the fourth moments of the increments of the process, specifically $E[|Z(T)(a) - Z(T)(b)|^4]$ $E [∣ Z (T) (a) - Z (T) (b) ∣^{4}]$ .
- Dirichlet Sum Approximation: The process is decomposed into a main term (a Dirichlet sum involving the von Mangoldt function $\Lambda_x(n)$ ) and an error term involving the zeros of the zeta function.
- Moment Bounds: The fourth moments of the Dirichlet sum are bounded using Lemma 3.2, which relies on the Montgomery-Vaughan identity.
- Handling Zeros and Error Terms: The contribution of the zeta zeros and the error term $e_x(s)$ is controlled by defining a "bad" set of times $Y_\epsilon$ (related to the proximity of $\tau$ to the imaginary parts of zeta zeros). By restricting $\tau$ to the complement of this set (which has measure $O(\epsilon T)$ ), the author bounds the error terms using estimates on the density of zeros and the specific choice of the smoothing function $\Lambda_x(n)$ .

Key Contributions and Results
The primary result is Theorem 1.1, which states that the normalized process
$Z(T)(\alpha) = \frac{1}{\sqrt{\log \log T}} \log \zeta\left(\frac{1}{2} + \frac{1}{(\log T)^\alpha} + i\tau\right)$
converges in law to a standard complex Brownian motion $B$ as $T \to \infty$ , for $\alpha \in [0, 1]$ .

This functional convergence yields several novel corollaries regarding the global behavior of $\zeta(s)$ :

Reflection Principle Analogue (Corollary 1.3): The maximum of $\log |\zeta(\sigma + i\tau)|$ over horizontal intervals, normalized by $\sqrt{\log \log T}$ , converges in distribution to the absolute value of a standard normal variable $|N(0, 1/2)|$ . This mirrors the reflection principle for Brownian motion.
Arcsine Law (Corollary 1.4): The distribution of the logarithmic measure of the set of $\sigma$ where $|\zeta(\sigma + i\tau)| \geq 1$ converges to the arcsine law.
Law of the Iterated Logarithm (Corollary 1.5): A result quantifying the abnormally large fluctuations of the maximum of $\log |\zeta|$ as one approaches the critical line horizontally, analogous to the law of the iterated logarithm for Brownian motion.
Local Time (Corollary 1.6 & 1.7): The paper establishes the weak convergence of the local time of $\text{Re} Z(T)$ to that of Brownian motion, implying that for large $T$ , $\log |\zeta|$ changes sign frequently within horizontal intervals near the critical line.

Significance
The paper claims that while the finite-dimensional convergence of $\log \zeta$ to a Gaussian vector was already understood (via methods in [4]), the tightness of the sequence of random functions was not a priori obvious and constitutes the main contribution. Establishing this tightness is necessary to deduce the corollaries regarding maxima, sign changes, and occupation measures.

The work extends the analogy between the Riemann zeta function and Random Matrix Theory (specifically the Gaussian Unitary Ensemble) from local statistics (pair correlation) and extreme value statistics (vertical intervals) to functional statistics along horizontal paths. The author notes that these corollaries are, to their knowledge, novel. The paper also draws a parallel to random unitary matrices, suggesting that a similar functional convergence to Brownian motion likely holds for the logarithm of the characteristic polynomial of random unitary matrices, though this is presented as a conjecture rather than a proven result within the text.

The methodology relies on standard tools from probabilistic number theory (Selberg's CLT techniques, Dirichlet series approximations, and zero-density estimates) but applies them in a novel functional framework to establish the Brownian nature of the zeta function's horizontal excursions.