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Smoothed Shifted Convolutions of Generalised Divisor Functions

The paper establishes an asymptotic formula for the smoothed shifted convolution of the generalised divisor function $d_k(n)$ and the divisor function $d(n)$ for $k \ge 4$ , featuring a power-saving error term with an exponent independent of $k$ that improves upon Topacogullari's 2018 result for sufficiently large $k$ .

Original authors: Cheuk Fung Lau

Published 2026-07-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Cheuk Fung Lau

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 number line as a vast, bustling city where every building is a number. Some buildings are special: the "prime" buildings, which can't be broken down into smaller blocks. To study how these prime buildings are arranged, mathematicians use a special tool called the "von Mangoldt function," which acts like a high-powered spotlight that only shines on the prime buildings.

But sometimes, we want to know about the "divisor" buildings. Every number has a certain number of ways it can be built from smaller blocks. For example, the number 6 can be built as $1 \times 6$ , $2 \times 3$ , $3 \times 2$ , or $6 \times 1$ . That's 4 ways. The "divisor function," $d(n)$ , counts these ways. The "generalized divisor function," $d_k(n)$ , is a more complex version that counts ways to build a number using exactly $k$ blocks.

The Big Puzzle: The Shifted Convolution

The paper tackles a specific, tricky puzzle: What happens when we look at two buildings, $n$ and $n+h$ , that are close to each other (separated by a distance $h$ ), and count how many ways both can be built?

Mathematicians call this a "shifted convolution." It's like asking: "If I pick a random house, and then look at the house $h$ doors down, how likely is it that both houses have a very specific number of ways to be built?"

For a long time, mathematicians could solve this puzzle for small numbers of blocks ( $k=2$ or $k=3$ ). But when the number of blocks gets large ( $k \ge 4$ ), the math gets incredibly messy. Previous attempts to solve it for large $k$ produced a "fuzziness" (an error term) that got worse and worse as $k$ increased. It was like trying to hear a whisper in a storm; the louder the storm (the larger $k$ ), the harder it was to hear the whisper.

The New Discovery: A Clearer Signal

Cheuk Fung (Joshua) Lau, the author of this paper, has found a new way to listen to that whisper. He proves a new formula that predicts the number of ways these paired buildings can be built, but with a crucial improvement: the "fuzziness" or error in his prediction does not get worse as $k$ gets larger.

Think of it like a radio. Previous radios had a static noise that grew louder the more you turned up the volume (increasing $k$ ). Lau has built a new radio where the static stays at a manageable level, no matter how high you turn the volume. This means his formula works much better for large values of $k$ than the previous best attempts (specifically, it improves on a result by Topacogullari from 2018).

How They Did It: The Detective Work

To solve this, Lau didn't just guess. He used a clever strategy involving "smoothing." Instead of counting every single building in a rigid, blocky way, he used a "smooth" function (a gentle curve) to weigh the buildings. This is like looking at the city through a slightly foggy lens that blurs the edges just enough to make the big patterns pop out, rather than getting stuck on the tiny details of every single brick.

He broke the problem down into smaller pieces, looking at how the buildings were grouped. He used a powerful new tool from a 2024 paper by Grimmelt and Merikoski, which acts like a master key for unlocking complex patterns in number theory. By combining this key with a technique called "Cauchy-Schwarz" (a way of comparing two lists of numbers to find their relationship), he was able to glue the pieces together without the error term exploding.

What They Proved (and What They Didn't)

The paper proves (it is a mathematical certainty, not just a guess) that for any large enough number $x$ , and for a shift $h$ that isn't too huge (specifically, $|h|$ must be smaller than roughly $x^{25/28}$ ), the number of ways to build these paired numbers follows a predictable pattern.

The formula looks like this:
$\text{Count} \approx \text{Main Pattern} + \text{Small Error}$

The "Main Pattern" is a polynomial (a fancy algebraic expression) that depends on $k$ , $h$ , and the smoothing function. The "Small Error" is the part that used to be a problem. Lau proves this error is roughly proportional to $x^{1 - \frac{7}{128}\eta}$ (where $\eta$ is a small number related to how big $h$ is).

Crucial Limits

It is important to note what this paper does not do. The author explicitly states that a "fixed power saving" for the "sharp cutoff" problem (counting without the smooth blur) is currently out of reach. In other words, while they can solve the puzzle with the "foggy lens" (smoothed version), they cannot yet prove the same result if you try to look at the buildings with perfect, sharp focus. The paper does not claim to have solved the hardest version of the problem, only a slightly softer, more manageable version that still yields a massive improvement for large $k$ .

The Bottom Line

This paper is a solid mathematical proof that we can now predict the behavior of these complex divisor pairs for large numbers of blocks with a level of precision that doesn't degrade as the numbers get bigger. It's a significant step forward in understanding the hidden rhythms of prime numbers and their neighbors, showing that even in the chaotic city of numbers, there are patterns that remain clear, no matter how loud the noise gets.

Technical Summary: Smoothed Shifted Convolutions of Generalised Divisor Functions

Problem Statement
The paper addresses the asymptotic evaluation of smoothed shifted convolution sums involving the generalised divisor function $d_k(n)$ and the standard divisor function $d(n)$ . Specifically, the author investigates sums of the form:
$\sum_{n} w\left(\frac{n}{x}\right) d_k(n) d(n+h)$
where $w$ is a smooth, compactly supported weight function, $k \ge 4$ is an integer, and $h$ is a non-zero integer shift. The goal is to derive an asymptotic formula with a main term and an error term that exhibits "power-saving" (i.e., the error is $O(x^{1-\delta})$ for some $\delta > 0$ ).

Previous results, notably by Topacogullari (2018), provided such formulas but with error exponents that deteriorated as $k$ increased. The central challenge addressed in this work is to obtain an asymptotic formula where the exponent of the power-saving error term is uniform in $k$ , thereby improving upon existing bounds for sufficiently large $k$ .

Methodology
The proof strategy follows a decomposition of the generalised divisor function $d_k(n)$ into its constituent factors $a_1 \cdots a_k = n$ . The methodology proceeds through several key stages:

Decomposition and Dyadic Splitting: The sum is decomposed based on the dyadic sizes of the variables $a_i$ . The author employs a Cauchy-Schwarz inequality to reduce the problem to estimating sums of the form:
$\sum_{n} w_1\left(\frac{r_1 n}{x}\right) w_2\left(\frac{r_2 n}{x}\right) d(r_1 n + h) d(r_2 n + h)$
where $r_1, r_2$ are square-free integers.
Spectral Methods and Determinant Equations: Instead of relying on classical sums of Kloosterman sums (as used in previous works), the paper utilizes a recent result by Grimmelt and Merikoski (2024). This involves rewriting the divisor sums as counting solutions to determinant equations of the form $ad - bc = \text{const}$ , twisted by periodic weights. The problem is framed within the context of automorphic forms on $\Gamma \backslash SL_2(\mathbb{Z})$ , where $\Gamma$ is a specific congruence subgroup.
Handling Arithmetic Obstructions: A significant portion of the technical work (Section 5) is dedicated to relaxing the coprimality conditions required by the spectral input. The author develops an iterative procedure using inclusion-exclusion principles to handle cases where the shift $h$ shares common factors with the coefficients $r_1, r_2$ . This allows the application of the spectral theorem even when $\gcd(h, r_1 r_2) \neq 1$ .
Case Analysis via Partitioning: To achieve uniformity in $k$ , the author proves a combinatorial lemma (Lemma 5.2) which partitions the possible sizes of the variables $A_1, \dots, A_k$ into three admissible cases. For each case, a specific error bound derived from the spectral analysis is applied. By carefully balancing these bounds, the author ensures that the resulting error exponent does not depend on $k$ .

Key Contributions and Results

Theorem 1.2 (General Asymptotic Formula): The paper establishes a general asymptotic formula for the smoothed shifted convolution. The error term is given by:
$O_{k,w,\varepsilon,\delta}\left( x^{1-\delta + 2\delta\theta + \varepsilon} \left(1 + \frac{|h|^{1/4}}{x^{1/4 - \delta/2}}\right) + x^{1-\delta + \theta/2 + \varepsilon} \left(1 + \frac{|h|^{\theta/2}}{x^{\theta/3 + 2\delta/3\theta}}\right) \right)$
where $\theta$ is the bound towards the Ramanujan-Selberg conjecture (currently $\theta = 7/64$ ). Crucially, the exponent of the power saving is independent of $k$ .
Theorem 1.1 (Main Result): By optimizing the parameters in Theorem 1.2, the author proves that for $k \ge 4$ and $|h| \ll x^{25/28 - \eta}$ , the sum satisfies:
$\sum_{n} w\left(\frac{n}{x}\right) d_k(n) d(n+h) = x P_{k,h,w}(\log x) + O_{k,w,\varepsilon}\left( x^{1 - \frac{7}{128}\eta + \varepsilon} + x^{\frac{127}{128} + \varepsilon} \right)$
Here, $P_{k,h,w}$ is a polynomial of degree $k$ .
Comparison with Prior Work: The paper demonstrates that for sufficiently large $k$ , the error exponent $127/128 \approx 0.992$ (or the $\eta$ -dependent term) is superior to the exponent derived by Topacogullari (2018), which worsens as $k$ increases. Specifically, the paper notes that the range of validity for $h$ extends up to $x^{25/28}$ , improving upon the $x^{15/19}$ range in Topacogullari's smoothed result.

Significance and Claims
The paper claims that its primary contribution is the uniformity of the error term with respect to $k$ . While fixed power-saving results for the sharp cutoff problem remain out of reach (as the existence of a uniform asymptotic for $\sum_{n \le x} d_k(n)$ is not yet known), this work successfully extends the smoothed shifted convolution results to a regime where the error term does not degrade as the generalised divisor function becomes more complex ( $k \to \infty$ ).

The author explicitly states that the improvement relies on two methodological shifts:

Replacing classical Kloosterman sum estimates with the spectral methods of Grimmelt and Merikoski (2024) regarding determinant equations.
Utilizing a more efficient "gluing" of variables via Cauchy-Schwarz, which allows for the specific partitioning of cases that yields the uniform bound.

The results are presented as a corollary of the general spectral estimates, offering a refined understanding of the distribution of divisor functions in shifted convolution sums for high-order generalisations.

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