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Local Moments of Möbius Fourier Polynomials and the Riemann Hypothesis

This paper establishes a local probabilistic reformulation of the Riemann Hypothesis by proving that the hypothesis is equivalent to the subpolynomial growth of arbitrarily high finite local moments of normalized Möbius Fourier polynomials evaluated at random points within a critical-scale arc.

Original authors: Alberto Verjovsky

Published 2026-07-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Alberto Verjovsky

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

The Great Number Hunt and the Whispering Circle

Imagine you are a detective trying to solve the ultimate mystery of mathematics: the Riemann Hypothesis. This isn't just a puzzle about numbers; it's a quest to understand the hidden rhythm of prime numbers, the building blocks of arithmetic that appear in everything from computer encryption to the structure of the universe. For over 160 years, mathematicians have been trying to prove that these primes follow a specific, predictable pattern, but the proof has remained elusive.

To get a handle on this mystery, the paper we are about to explore uses a few key ideas. First, there's the Möbius function, which acts like a cosmic switch for numbers, assigning them values of +1, -1, or 0 based on their factors. If you add up these switches for all numbers up to a certain point, you get the Mertens function. The Riemann Hypothesis is essentially a promise that this sum doesn't grow too fast; it stays within a "square-root" boundary.

Second, the paper uses Fourier polynomials. Think of these as musical instruments. Just as a complex sound can be broken down into simple notes (frequencies), a complex pattern of numbers can be turned into a wave made of different musical tones. The "Riemann Hypothesis" in this context is a claim about how loud or quiet this musical wave gets.

Finally, the paper introduces a bit of probability. Instead of looking at the wave at just one fixed spot, imagine spinning a wheel to pick a random spot on the wave to measure. The big question is: if the Riemann Hypothesis is true, what does the "volume" of this wave look like when we take random samples from a tiny, shrinking slice of the circle?

The Paper's Discovery: Listening to the Whisper

In this paper, mathematician Alberto Verjovsky proposes a clever new way to test the Riemann Hypothesis. He doesn't try to prove it directly. Instead, he creates a "local probabilistic reformulation." Think of it like this: imagine you have a giant, invisible drum (the Möbius polynomial) that vibrates with a complex rhythm. The Riemann Hypothesis claims that this drum never gets too loud.

Verjovsky's idea is to stop listening to the whole drum and instead focus on a tiny, shrinking patch of its surface. He picks a random point on this tiny patch—specifically, a patch so small it's only about $1/N$ wide, where $N$ is the number of notes in the song. He then asks: "If I measure the loudness (the 'moment') of the drum at this random spot, over and over again, how does the average loudness behave?"

The paper proves a fascinating equivalence: The Riemann Hypothesis is true if and only if the loudness of these random samples stays "subpolynomial."

What does "subpolynomial" mean in plain English? It means the loudness doesn't explode. If the Riemann Hypothesis is true, even if you pick a very high "volume setting" (a high mathematical power $q$ ) to measure the sound, the average noise level on that tiny patch will grow incredibly slowly—so slowly that it's almost flat. It won't shoot up like a rocket; it will barely creep up at all.

The paper argues that if the Riemann Hypothesis were false, the drum would have a massive spike right at the center (the origin). Because the drum is made of waves, a massive spike at the center would force the sound to stay loud for a little while on the surrounding area. By measuring the "moments" (the average loudness) on that tiny, shrinking arc, Verjovsky shows you can detect that spike. If the average loudness stays low (subpolynomial) no matter how high you set your volume meter, then there is no spike, and the Riemann Hypothesis holds.

How the Detective Works

The paper builds a bridge between two worlds: the deterministic world of number theory and the random world of probability.

The Setup: The author defines a random variable, $X_{N,c}$ , which is the value of the Möbius polynomial at a point chosen uniformly at random from a tiny interval of radius $c/N$ .
The Tool: He uses a "local moment-to-point-value inequality." This is a mathematical rule that says: "If you know the average loudness of a wave on a tiny patch, you can figure out how loud the wave is right at the center."
The Result: The paper demonstrates that if the Riemann Hypothesis is true, then for any fixed size of the patch (as long as it shrinks at the rate of $1/N$ ), the average loudness of these random samples will grow slower than any power of $N$ . In other words, $M_{q,c}(N) = O_{\eta,q,c}(N^\eta)$ for every tiny $\eta > 0$ .

Crucially, the paper clarifies what this is not. It is not a new method to prove the Riemann Hypothesis from scratch. It doesn't randomize the numbers themselves (like the old "Denjoy heuristic" which pretended the numbers were random coins). Instead, it keeps the numbers exactly as they are (deterministic) and only randomizes where we look.

The paper also rules out the idea that a single measurement is enough. It shows that you need to look at "arbitrarily high finite moments." If you only check the average loudness (a low moment), you might miss a sharp spike. But if you crank up the sensitivity to measure the "extreme" loudness (high moments), you can't hide a spike anymore. The paper proves that the Riemann Hypothesis is equivalent to the statement that no matter how high you crank the sensitivity, the local loudness stays under control.

Why This Matters

This approach is like finding a new way to listen to a whisper. Instead of trying to hear the whisper across a noisy stadium (the whole circle), the author suggests listening to a tiny, quiet corner where the whisper is most likely to be heard. If the whisper is truly quiet in that corner, no matter how hard you strain to hear it, then the Riemann Hypothesis is true.

The paper concludes that this "critical-scale local criterion" complements existing theories. It connects the ancient problem of prime numbers with modern ideas about random waves and local concentration. While it doesn't solve the Riemann Hypothesis today, it gives mathematicians a precise, probabilistic language to describe what the solution must look like: a world where the local fluctuations of these number-waves remain perfectly tame, even on the smallest scales.

Technical Summary: Local Moments of Möbius Fourier Polynomials and the Riemann Hypothesis

Problem Statement
The paper addresses the Riemann Hypothesis (RH) through the lens of the Mertens summatory function, $M(x) = \sum_{n \le x} \mu(n)$ . A classical equivalent of RH is the bound $M(x) = O_\epsilon(x^{1/2+\epsilon})$ for every $\epsilon > 0$ . While probabilistic heuristics (such as Denjoy's) treat the Möbius function $\mu(n)$ as independent random signs to explain the square-root cancellation inherent in RH, these models discard the deterministic multiplicative structure of the coefficients. The paper seeks a reformulation that retains the deterministic arithmetic information of $\mu(n)$ while introducing randomness in a way that allows for a probabilistic analysis of the function's behavior.

Methodology
The author defines the normalized Möbius Fourier polynomial:
$P_N(t) = \frac{1}{\sqrt{N}} \sum_{n \le N} \mu(n) e^{2\pi i n t}, \quad t \in \mathbb{T} = \mathbb{R}/\mathbb{Z}.$
Instead of randomizing the coefficients $\mu(n)$ , the methodology introduces randomness by evaluating $P_N(t)$ at a random point $U_{N,c}$ chosen uniformly from a shrinking arc $I_{N,c} = [-c/N, c/N]$ , where $c > 0$ is a fixed constant. The scale $N^{-1}$ is chosen because it corresponds to the period of the highest frequency mode ( $n=N$ ) in the polynomial; on this scale, the polynomial is not effectively constant, yet the interval is small enough to be considered "local."

The core analytical tool is a local moment-to-point-value inequality. The paper establishes that the value of a trigonometric polynomial at the origin, $|S_N(0)|$ , can be bounded by its local $L^q$ moments on an interval of radius $r \sim N^{-1}$ . Specifically, for the Möbius polynomial, the paper derives an inequality linking the Mertens function $M(N)$ to the local moments $M_{q,c}(N) = (\mathbb{E}|P_N(U_{N,c})|^q)^{1/q}$ :
$|M(N)| \le C_{q,c} \max \left\{ \sqrt{N} M_{q,c}(N), \, N^{\frac{1}{2} + \frac{1}{2(q+1)}} M_{q,c}(N)^{\frac{q}{q+1}} \right\}.$

Key Contributions and Results
The primary contribution is Theorem 1.3, which provides an equivalent probabilistic formulation of the Riemann Hypothesis:

Equivalence: The Riemann Hypothesis holds if and only if, for every $\eta > 0$ and every finite $q \ge 1$ , the local moments satisfy the subpolynomial growth condition:
$M_{q,c}(N) = O_{\eta, q, c}(N^\eta).$
Weakened Condition: The hypothesis also holds if, for every $\eta > 0$ , there exists an unbounded set of exponents $Q_\eta \subseteq [1, \infty)$ such that the subpolynomial bound holds for all $q \in Q_\eta$ .
Mechanism of Recovery: The paper demonstrates that arbitrarily high finite moments are required to recover the critical exponent $1/2$ in the Mertens bound. A single fixed $q$ yields a bound of $O(N^{1/2 + \epsilon})$ with a loss term $1/(2(q+1))$ that vanishes only as $q \to \infty$ . Thus, the "local concentration" of the polynomial's mass on the critical scale $N^{-1}$ encodes the global arithmetic information of $M(N)$ .

The paper also contrasts this local criterion with global $L^p$ -semiflatness criteria (studied by el Abdalaoui) and discusses the naturalness of the $N^{-1}$ scale. It notes that while RH implies subpolynomial bounds on arcs of size $r_N = N^{-1+o(1)}$ , determining the maximal shrinking scale for which this holds remains an open question (Question 5.3).

Significance and Claims
The author explicitly states that this work is not a new approach to proving the Riemann Hypothesis. Rather, its significance lies in:

Providing an equivalent formulation of RH that isolates the role of local concentration on the critical scale determined by the highest frequency.
Offering a quantitative local principle showing how high local moments on a shrinking arc recover the distinguished value $P_N(0) = M(N)/\sqrt{N}$ .
Complementing existing frameworks: It bridges Denjoy's random-walk heuristic (which randomizes coefficients) and Kahane's theory of random Fourier series by retaining deterministic coefficients while randomizing the evaluation point.
Establishing a "critical-scale local criterion" that places the classical Mertens criterion in a form comparable to probabilistic flatness questions for Fourier polynomials.

The paper concludes that the behavior of these local random variables—specifically, whether they exhibit subpolynomial moments of arbitrarily high finite order—serves as a precise probabilistic proxy for the validity of the Riemann Hypothesis.

The Great Number Hunt and the Whispering Circle

The Paper's Discovery: Listening to the Whisper

How the Detective Works

Why This Matters

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