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Character sums over smooth numbers

This paper establishes that the average magnitude of character sums over $y$ -smooth numbers is significantly smaller than the square root of the count of such numbers, specifically achieving an $o(\sqrt{\Psi(x,y)})$ bound when the modulus $q$ is sufficiently large relative to $x$ and $y$ falls within a specific intermediate range.

Original authors: Seth Hardy, Max Wenqiang Xu

Published 2026-07-02

📖 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: Finding Order in a Chaotic Crowd

Imagine you are at a massive, chaotic music festival. There are millions of people (numbers) wandering around. Some people are wearing simple, plain clothes (numbers with small prime factors), while others are wearing wild, complex outfits with many layers (numbers with large prime factors).

In mathematics, there is a special group of people called "smooth numbers." These are the people whose outfits are made entirely of small, simple patterns (their prime factors are all smaller than a certain limit, $y$ ).

The authors of this paper are trying to solve a puzzle about how these smooth numbers behave when they are "sung" by different choirs. In math terms, these choirs are called Dirichlet characters. Each choir sings a different tune (assigns a different value) to every number.

The question the authors ask is: If we listen to all the choirs at once, how much does the noise cancel out?

Usually, if you have a huge crowd and everyone is singing randomly, the noise is loud. But if the singers are perfectly coordinated (orthogonal), the noise cancels out, and the total volume drops significantly. The authors wanted to prove that for "smooth numbers," the noise canc out even better than we previously thought, but only under specific conditions.

The Cast of Characters

The Smooth Numbers ( $\Psi(x, y)$ ): Think of these as the "easy" numbers. Just like a smooth stone has no sharp edges, these numbers have no "sharp" large prime factors. They are built entirely from small building blocks.
The Choirs (Dirichlet Characters): These are the mathematical functions that assign values to numbers. Imagine a choir where every member sings a different note for every number they see.
The "Random" Singer (Steinhaus Random Multiplicative Function): Before this paper, mathematicians had a very helpful "imaginary" singer. This singer is completely random but follows strict rules. By studying this imaginary singer, mathematicians could guess how the real choirs behave. The authors used this imaginary singer as a blueprint to build their proof for the real choirs.

The Main Discovery: A "Magic" Cancellation

The paper proves a specific result: When you sum up the songs of all the choirs for these smooth numbers, the total volume is much quieter than the "worst-case scenario" predicts.

In the old days, mathematicians used a rule of thumb (the Cauchy-Schwarz inequality) that said, "The noise will be loud, roughly the square root of the number of people."

The Old Prediction: If there are 1,000 smooth numbers, the noise might be around $\sqrt{1000} \approx 31$ .
The New Discovery: The authors show that the noise is actually much quieter than 31. It's like finding out that instead of a roar, the crowd is actually whispering.

However, there is a catch (The "Condition"):
This magic cancellation only happens if the choir is huge compared to the crowd. Specifically, the number of choirs ( $q$ ) must be significantly larger than the number of people ( $x$ ).

Analogy: Imagine trying to find a pattern in a small group of 10 people. It's hard to tell if they are random or coordinated. But if you have a stadium of 1,000,000 choirs, you can clearly see that they are canceling each other out perfectly. The paper proves that if the choir is big enough (specifically, if $q$ is slightly larger than $x$ ), the cancellation is real and substantial.

How They Did It: The "Truncated" Recipe

The authors didn't just guess; they built a mathematical machine to prove it. Here is the simplified version of their method:

The Problem: The real choirs are tricky because the singers aren't truly independent (unlike the imaginary random singer). You can't just multiply their probabilities together easily.
The Solution (The Taylor Expansion): The authors decided to look at the "recipe" for the noise. Instead of trying to analyze the whole infinite song, they chopped it up into a short list of ingredients (a truncated series).
The "Perfect" Cancellation: They realized that if they only looked at the first few ingredients (the first few prime factors), the math becomes much simpler. Because the choir is so huge, the "ingredients" in this short list behave almost perfectly independently.
The Safety Net: They proved that the parts of the song they didn't include (the long tail of the recipe) are so quiet that they don't matter. They used a technique called "Rankin's trick" (think of it as a mathematical safety net) to prove that ignoring the long tail doesn't change the result.

The "Saving" Factor

The paper introduces a "Saving Factor" ( $S$ ). Think of this as a discount coupon.

Without the coupon: You pay the full price (the square root bound).
With the coupon: You pay a tiny fraction of the price.
The size of the discount depends on how "smooth" the numbers are and how big the choir is. If the choir is huge and the numbers are very smooth, the discount is massive.

Why This Matters (In the Paper's Context)

The authors note that this is the first time anyone has successfully proven that this "quieting effect" happens for smooth numbers in this specific way.

They compared their result to the "imaginary random singer" (which is known to be very quiet) and showed that the real choirs behave almost as well as the imaginary one, provided the choir is big enough.
They also showed that this works even if you twist the songs with other mathematical functions (like the Liouville function), making the result very robust.

Summary in One Sentence

The authors proved that if you have a massive choir of mathematical singers, and you ask them to sing only about "smooth" numbers (numbers made of small building blocks), the noise they make will cancel out almost perfectly, leaving a result much quieter than anyone expected—provided the choir is large enough to make the math work.

Technical Summary: Character Sums Over Smooth Numbers

Problem Statement
This paper addresses the problem of bounding the first absolute moment of partial sums of Dirichlet characters (and continuous characters) restricted to $y$ -smooth numbers. Let $\Psi(x, y)$ denote the count of integers $n \le x$ such that all prime factors of $n$ are at most $y$ (i.e., $P(n) \le y$ ). The authors investigate the quantity:
$\frac{1}{\phi(q)} \sum_{\chi \bmod q} \left| \sum_{\substack{n \le x \\ P(n) \le y}} \chi(n) \right|$
The classical "trivial bound" for this sum, derived from the Cauchy–Schwarz inequality and orthogonality of characters, is of order $\sqrt{\Psi(x, y)}$ . The central question is whether one can establish a non-trivial saving over this square-root bound, particularly in ranges where $y$ is small relative to $x$ (the "smoothness" parameter).

Methodology
The authors adapt techniques from the theory of random multiplicative functions, specifically the work of Harper and their own previous results on Steinhaus random multiplicative functions, to the deterministic setting of Dirichlet characters.

Perron's Formula and Saddle Point: The proof utilizes Perron's formula to express the partial sum as a contour integral. The contour is shifted to the line $\text{Re}(s) = \alpha(x, y)/2$ , where $\alpha(x, y)$ is the saddle point defined by $\sum_{p \le y} \frac{\log p}{p^{\alpha} - 1} = \log x$ .
Truncated Taylor Expansion: A key difficulty in the deterministic setting is that the values $\chi(p)$ for $p \le y$ are not independent, preventing the direct factorization of the associated $L$ -function $L(s, \chi, y)$ into a product of expectations. To circumvent this, the authors expand the logarithm of the $L$ -function into a Taylor series. They truncate this expansion at a level $K$ (where $K \approx \log x / \log y$ ) to create a generating function involving sums over primes.
Orthogonality and Truncation: The truncation level $K$ is chosen such that $y^K \le q$ . This condition ensures that the orthogonality relations for Dirichlet characters modulo $q$ can be applied "perfectly" to the truncated sums, allowing for the calculation of moments.
Two Key Lemmas: The proof relies on two main lemmas:
- Lemma 1 (Approximation): It establishes that the truncated Taylor expansion approximates the original sum (weighted by $n^{-\alpha/2-it}$ ) with a small error. The error term accounts for integers with high prime factor multiplicity or those with more than $K$ prime factors.
- Lemma 2 (Tail Bound): It provides a tail bound for the sum over primes $P(\alpha/2 + it, \chi, y)$ . This lemma demonstrates that for almost all characters, this sum is small (of order $\sqrt{u}$ , where $u = \log x / \log y$ ), utilizing moment bounds similar to those used for the Riemann zeta function.

Key Contributions and Results
The paper establishes the following main result (Theorem 1):

Theorem 1: Let $(\log x)^6 \le y \le x^{\frac{1}{32 \log \log x}}$ and $q \ge x^{1+\varepsilon}$ for some small quantifiable $\varepsilon > 0$ . Then:
$\frac{1}{\phi(q)} \sum_{\chi \bmod q} \left| \sum_{\substack{n \le x \\ P(n) \le y}} \chi(n) \right| \ll \sqrt{\Psi(x, y)} S(x, y, q)$
where $S(x, y, q)$ is a saving factor defined as:
$S(x, y, q) = (\log x)^{3/2}(\log y)^{1/2} \left( y^{1/2 - \alpha(x, y)} + \exp\left\{ -\frac{u}{4} \left( \frac{\log \min\{q, x^{3/2}\}}{\log x} - 1 \right)^2 \right\} \right)$
The authors prove that $S(x, y, q) = o(1)$ uniformly in the specified range, provided $\frac{\log q}{\log x}$ is sufficiently large (specifically $> 1 + \frac{\sqrt{6 \log \log x + 2 \log \log y + \eta(x)}}{u}$ ).

Generalizations:

Theorem 2: The result extends to sums twisted by completely multiplicative functions $h(n)$ with $|h(p)| \le 1$ , such as the Liouville function $\lambda(n)$ .
Theorem 3: The results are generalized to continuous characters ( $n^{it}$ ) averaged over $t \in [T, 2T]$ , and to averages over primitive Dirichlet characters with conductors up to $Q$ .

Significance and Claims
The authors claim that Theorem 1 is the first result in the literature to provide non-trivial insight into the first absolute moment of character sums over smooth numbers.

Comparison to Random Models: The result mirrors the behavior found in random multiplicative functions (where the expectation is $o(\sqrt{\Psi(x, y)})$ ), but the saving factor here is weaker than in the random case. The authors attribute this to the lack of genuine independence among $\chi(p)$ values, which forces a reliance on orthogonality that requires $q$ to be slightly larger than $x$ ( $q \ge x^{1+\varepsilon}$ ).
Range of Validity: The saving is substantial when $\varepsilon$ is fixed away from zero. However, the authors note that the theorem yields no improvement over the trivial Cauchy–Schwarz bound if $q$ is too close to $x$ (specifically if $\frac{\log q}{\log x} \le 1 + O(u^{-1/2})$ ). This limitation is intrinsic to the proof strategy, which relies on the "anatomy" of $y$ -smooth numbers requiring $q$ to be large enough to capture the typical number of prime factors ( $u$ ) via orthogonality.
Modesty: The paper explicitly states that the bound obtained is weaker than the bound in the random multiplicative case (which is believed to be close to optimal) and that the proof does not directly compare the problem to the random multiplicative case, but rather adapts the strategy. The authors acknowledge that proving similar results for the Möbius function introduces technical difficulties regarding sign changes that are not resolved in this work.

In summary, the paper successfully transfers the "saddle point + truncated expansion + orthogonality" strategy from the probabilistic setting to the deterministic setting of Dirichlet characters, achieving a non-trivial saving over the square-root bound for smooth numbers under specific conditions on the modulus $q$ .