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Helson's conjecture for smooth numbers

The paper proves that for a Steinhaus random multiplicative function, the partial sums over $y$ -smooth numbers exhibit strictly better than square-root cancellation uniformly across the entire range $2 \leq y \leq x$ , thereby establishing a quantitative version of Helson's conjecture for smooth numbers.

Original authors: Seth Hardy, Max Wenqiang Xu

Published 2026-02-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Seth Hardy, Max Wenqiang Xu

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 Big Picture: The Great Number Shuffle

Imagine you have a giant bag of numbers, from 1 up to a very large number $x$ . Inside this bag, every number has a "personality" assigned to it by a random generator. This generator is called a Steinhaus random multiplicative function.

Think of this generator like a magical coin flip for every prime number (2, 3, 5, 7, etc.).

For the number 2, the coin lands on a random point on a circle (like a clock hand pointing anywhere).
For the number 3, it lands on another random point.
For any other number, its personality is just the product of the personalities of the prime numbers that make it up. (For example, if 6 is made of 2 and 3, its personality is the result of multiplying the "coin flips" of 2 and 3).

The mathematicians in this paper are asking a simple question: If you add up all these random personalities from 1 to $x$ , do they cancel each other out?

In a normal random walk (like a drunk person stumbling left and right), you expect the total distance from the start to be roughly the square root of the number of steps. This is called "square-root cancellation." It's the standard rule of thumb for randomness.

Helson's Conjecture (proven true for the full range of numbers by a mathematician named Harper in 2022) suggested that these specific number personalities are extra random. They cancel out better than the standard square-root rule. The total sum is actually smaller than expected.

The New Twist: The "Smooth" Numbers

The authors of this paper, Seth Hardy and Max Wenqiang Xu, asked: Does this "super-cancellation" still happen if we only look at a specific subset of numbers?

They focused on "Smooth Numbers."

Analogy: Imagine a smooth number is like a smooth stone in a river. It has no sharp edges. In math terms, a "y-smooth" number is one whose prime factors are all small (less than or equal to $y$ $y$ ).
- If $y$ is small, the number is made of tiny building blocks (like 2, 3, 5).
- If $y$ is large (close to $x$ ), the number can be made of big blocks too.

The question is: If we only sum up the personalities of these "smooth" stones, do they still cancel out better than the square-root rule?

The Discovery: Yes, Everywhere!

The paper proves a surprising result: Yes, they always cancel out better than the square-root rule, no matter how small or large the "smoothness" limit ( $y$ ) is.

The authors didn't just say "yes"; they broke the problem down into three different "neighborhoods" of smoothness, each requiring a different tool to solve:

1. The "Moderately Smooth" Neighborhood (The Middle Ground)

The Scenario: The numbers are made of prime factors that are somewhat small, but not tiny.
The Analogy: Imagine a crowd of people where most are average height. You might expect the tallest and shortest people to balance out.
The Surprise: The authors found that the "average" behavior of these numbers is dominated by very unlikely events.
- Usually, the random personalities cancel out nicely.
- However, the mathematical "average" is secretly being pulled by rare, wild fluctuations where the numbers line up in a weird way.
- Because these wild events are so rare, the actual sum is much smaller than the "worst-case" estimate. It's like a casino where the house usually wins, but the mathematical average is skewed by one person winning the jackpot once in a million years. The authors proved that for this range, the "jackpot" events are so rare that the total sum is significantly smaller than expected.

2. The "Very Smooth" Neighborhood (Tiny Building Blocks)

The Scenario: The numbers are made of very small primes (like 2, 3, 5).
The Analogy: Imagine a tower built only of tiny Lego bricks.
The Result: Here, the randomness is so strong that the numbers cancel out incredibly well. The authors used a different mathematical trick (looking at the numbers near the "imaginary axis" in complex math) to show that the sum is tiny. It's like a chaotic storm that somehow settles down into perfect silence.

3. The "Almost Full" Neighborhood (The Edge Case)

The Scenario: The numbers are allowed to have almost any prime factor (so $y$ is very close to $x$ ).
The Analogy: This is the full bag of numbers again, but we are looking at the very edge.
The Result: This connects back to the original work by Harper. The authors showed that even here, the "super-cancellation" holds. They used a concept called Gaussian Multiplicative Chaos (a fancy way of describing how random waves interact). They proved that even though the math gets tricky near the edge, the waves still interfere with each other enough to make the total sum smaller than the square-root rule predicts.

Why Does This Matter? (According to the Paper)

The paper mentions one specific reason this is useful: Counting Smooth Numbers in Short Intervals.

The Problem: Mathematicians want to know: "If I look at a very short stretch of numbers (like from 1,000,000 to 1,000,100), will I definitely find a 'smooth' number?"
The Connection: The authors suggest that their proof of "super-cancellation" could help prove that smooth numbers appear more frequently in these short stretches than we currently know.
The Limit: They explicitly state that their result suggests a way to break a "square-root barrier" in counting these numbers. They do not claim to have solved the counting problem yet, but they have provided a powerful new tool (the proof of the cancellation) that could be used to solve it.

Summary

In simple terms, this paper is a tour de force of number theory. The authors took a famous rule about randomness (that sums usually cancel out to the square root of the count) and asked, "What if we only look at numbers made of small pieces?"

They proved that even in this restricted world, the randomness is even stronger than we thought. The numbers cancel out better than the standard rule, and they did this by showing that the "worst-case" scenarios are actually incredibly rare. They used three different mathematical "flashlights" to illuminate this phenomenon across the entire spectrum of smooth numbers, from the tiniest to the largest.

Technical Summary: Helson's Conjecture for Smooth Numbers

Problem Statement
The paper investigates the statistical behavior of partial sums of Steinhaus random multiplicative functions, denoted $f(n)$ , restricted to the set of $y$ -smooth numbers. A $y$ -smooth number is an integer $n$ whose largest prime factor $P(n)$ satisfies $P(n) \le y$ . Let $\Psi(x, y)$ denote the count of such numbers up to $x$ . The central question is whether the first absolute moment of these sums, $E\left| \sum_{n \le x, P(n) \le y} f(n) \right|$ , exhibits "better than square-root cancellation."

In the context of random multiplicative functions, the trivial bound derived from the Cauchy–Schwarz inequality is $\sqrt{\Psi(x, y)}$ . Helson's conjecture, resolved by Harper for the full sum ( $y=x$ ), posits that the expectation is actually $o(\sqrt{x})$ . This paper asks whether this phenomenon of enhanced cancellation persists uniformly across the entire range of smoothness parameters $2 \le y \le x$ , or if there exists a transition range where the cancellation degrades to the square-root level.

Methodology
The authors employ a combination of analytic number theory techniques and probabilistic arguments tailored to the specific structure of smooth numbers. The core methodology involves:

Perron's Formula and Euler Products: The partial sums are related to integrals of random Euler products $F_y(s) = \prod_{p \le y} (1 - f(p)p^{-s})^{-1}$ via Perron's formula. The expectation of the absolute sum is bounded by integrals involving $F_y(\sigma/2 + it)$ .
Saddle Point Analysis: The authors utilize the saddle point $\alpha = \alpha(x, y)$ , which minimizes the Rankin's trick upper bound for $\Psi(x, y)$ . The analysis shifts the integration contour to the line $\text{Re}(s) = \alpha/2$ (or $\alpha(x^2, y)/2$ in specific ranges).
Moment Estimates of Random Euler Products: A critical component is the evaluation of moments $E|F_y(\sigma/2 + it)|^\beta$ $E ∣ F_{y} (σ /2 + i t) ∣^{β}$ . Unlike the full sum case where the critical line $\text{Re}(s)=1/2$ $Re (s) = 1/2$ is central, this work analyzes Euler products shifted off the critical line.
- For "moderate" $y$ , the authors show that the first moment is significantly smaller than the square root of the second moment due to the dominance of "unlikely events" in the Gaussian approximation of $\log |F_y|$ .
- For $y$ close to $x$ , the authors adapt Harper's method involving Critical Gaussian Multiplicative Chaos (GMC). They isolate the contribution of small primes (where the process behaves like a log-correlated Gaussian field) and treat the remaining large primes via conditioning and Cauchy–Schwarz.
- For very small $y$ , the authors use specific estimates for Euler products near the imaginary axis, accounting for the high multiplicity of prime factors in smooth numbers.
Short Interval Estimates: The proof for large $y$ relies on Lemma 2.5, which provides bounds for sums of integers with restricted prime factors in short intervals, a result derived from Hildebrand's work.

Key Contributions and Results
The main result is Theorem 1.1, which establishes that for the entire range $2 \le y \le x$ :
$E\left| \sum_{1 \le n \le x, P(n) \le y} f(n) \right| = o\left( \Psi(x, y)^{1/2} \right).$
This confirms that "better than square-root cancellation" holds uniformly for $y$ -smooth numbers, regardless of how small $y$ is relative to $x$ . The paper provides quantitative bounds for three distinct regimes:

Moderate $y$ ( $C \log x \le y \le x^{1/\log \log x}$ ):
The authors prove a strong saving, showing the expectation is bounded by $\Psi(x, y)^{1/2} \exp(-u(\log 2 + o(1)))$ , where $u = \log x / \log y$ . This result (Theorem 1.2) is derived without relying on critical GMC phenomena. Instead, it exploits the fact that the dominant contribution to the Euler product moments comes from highly unlikely events where the log-Euler product is significantly larger than its typical maximum. This leads to a saving of at least a factor of $e^{cu}$ over the trivial bound.
Large $y$ ( $e^{(\log \log x)^2} \le y \le x$ ):
For $y$ close to $x$ , the paper proves (Theorem 1.3) that the expectation is bounded by $\Psi(x, y)^{1/2} (\log \frac{1}{1-\alpha})^{-1/4}$ . This recovers the $(\log \log x)^{-1/4}$ saving observed by Harper for the full sum. The proof adapts Harper's critical GMC argument by conditioning on small primes (where the GMC behavior holds) and handling the larger primes via short-interval estimates.
Small $y$ ( $2 \le y \le C \log x$ ):
For very small $y$ , the standard Gaussian approximations fail due to the high multiplicity of prime factors. The authors prove (Theorem 1.4) that cancellation still occurs, with bounds depending on the specific range of $y$ . For instance, when $y$ is bounded, the expectation is $O(\sqrt{\Psi(x, y)} \sqrt{\frac{\log \log x}{\log x}})$ .

Significance and Claims
The paper claims to resolve the question of whether the "better than square-root cancellation" phenomenon is universal for smooth numbers. The authors explicitly state that there is no transition range where this cancellation disappears; the enhancement holds uniformly across all $y$ .

The significance of the work lies in:

Uniformity: Extending Helson's conjecture to sets with non-trivial arithmetic structure (smooth numbers) across the full parameter range.
New Mechanisms: Identifying a distinct source of cancellation for moderate $y$ that does not rely on critical Gaussian multiplicative chaos. This mechanism arises from the specific variance structure of the Euler product when evaluated off the critical line, where the moments are controlled by rare, large deviations.
Potential Applications: The authors suggest that their results could be applied to the classical problem of counting $y$ -smooth numbers in short intervals $[x, x+h]$ . Specifically, they note that their quantitative bounds could potentially improve the known length of intervals $h$ required to guarantee the existence of $x^\varepsilon$ -smooth numbers, pushing the bound closer to $h \gg \sqrt{x}/(\log \log x)^{1/4}$ . However, they frame this as a potential consequence of their bounds rather than a proven application within the text.

The paper concludes that the "better than square-root" behavior is a robust feature of random multiplicative functions, persisting even when the support of the sum is restricted to smooth numbers of varying degrees of smoothness.