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Triple convolution sums of the generalised divisor functions and related sums over primes

This paper investigates the asymptotic behavior of triple convolution sums involving generalized divisor functions and their shifted variants over primes, providing a predicted asymptotic formula, deriving a lower bound of the correct order using several-variable Tauberian theorems, and establishing an explicit lower bound for the prime-restricted sum by combining Tauberian theory with the Bombieri-Vinogradov theorem.

Original authors: Bikram Misra, Biswajyoti Saha

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bikram Misra, Biswajyoti Saha

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 you are a detective trying to solve a mystery about numbers. Specifically, you are looking at a massive crowd of integers (1, 2, 3, 4...) and asking a very specific question: "How many ways can these numbers be broken down into smaller pieces, and how does that change if we look at numbers right next to them?"

This paper by Bikram Misra and Biswajyoti Saha is like a high-tech investigation into the hidden patterns of these "divisor" numbers. Here is the story of their discovery, broken down into simple concepts.

1. The Characters: The "Divisor" Family

First, let's meet the main characters.

The Divisor Function ( $d(n)$ ): Imagine $n$ is a Lego tower. The divisor function counts how many different ways you can break that tower apart into smaller, whole Lego blocks. For example, the number 6 can be broken into $1\times6$ , $2\times3$ , $3\times2$ , or $6\times1$ . So, $d(6) = 4$ .
The Generalized Divisor ( $d_k(n)$ ): This is a super-charged version. Instead of just breaking the tower into two pieces, you are breaking it into k pieces. If $k=3$ , you are asking: "How many ways can I write 6 as a product of three numbers?" ( $1\times1\times6$ , $1\times2\times3$ , etc.).

2. The Mystery: The Triple Convolution Sum

The authors are studying a specific "sum" (a giant addition problem). They are looking at three numbers that are close to each other: $n-h$ , $n$ , and $n+h$ (like three friends standing in a line).

They want to know: If I multiply the "divisor counts" of these three friends together and add up the results for everyone in the crowd, what is the total?

Mathematically, they are calculating:
$\text{Total} = \sum (\text{Divisors of } n+h) \times (\text{Divisors of } n) \times (\text{Divisors of } n-h)$

The Problem:
For a long time, mathematicians could easily predict the answer for two friends (a double sum). But for three friends (a triple sum), the math gets incredibly messy. It's like trying to predict the weather for three different cities at once; the variables interact in complex ways. No one had a solid proof for the exact answer for three friends until now.

3. The Prediction: The "Crystal Ball"

Before proving anything, the authors used a "Crystal Ball" (a combination of probability theory and advanced complex math) to predict what the answer should be.

They guessed that as the crowd gets huge (as $x$ goes to infinity), the total sum follows a very specific pattern:
$\text{Total} \approx \text{Constant} \times x \times (\log x)^{\text{power}}$

The "Constant" is a special number that depends on the gap between the friends ( $h$ ). It's like a "fingerprint" of the specific spacing between the numbers.

4. The Investigation: Two Tools for the Job

To prove their prediction was right, the authors used two powerful detective tools:

Tool A: The "Tauberian Theorem" (The Multi-Variable X-Ray)

This is a sophisticated mathematical technique that looks at the "shape" of the numbers in a multi-dimensional space.

The Analogy: Imagine you have a giant, invisible cloud of data. You can't see the individual drops, but you can see the overall shape of the cloud. The Tauberian theorem allows you to look at the "edges" of this cloud and deduce exactly how many drops are inside.
The Result: Using this tool, the authors proved that the total sum is at least as big as their prediction. They couldn't prove it was exactly the prediction yet (that's the hard part), but they proved it wasn't smaller. They successfully recovered the "fingerprint" (the constant) they predicted earlier.

Tool B: The "Prime Time" Investigation (The Titchmarsh Divisor Problem)

The authors then asked a harder question: What if we only look at the "Prime" numbers? (Primes are the atoms of math; numbers like 2, 3, 5, 7 that can't be broken down further).

The Challenge: Primes are rare and unpredictable. It's like trying to find a specific type of rare bird in a forest.
The Solution: They used a famous theorem called Bombieri-Vinogradov. Think of this as a "super-scope" that lets you see the distribution of primes clearly, even when they are far apart.
The Result: They proved a lower bound for the sum over primes, again matching their predicted "fingerprint."

5. The "Why" (The Probabilistic Route)

In the final section, the authors explain why their predicted constant makes sense using a fun analogy: Randomness.

Imagine every prime number is a coin flip.

For most numbers, the "divisor count" behaves randomly.
However, because the three numbers ( $n-h, n, n+h$ ) are linked, their "coin flips" aren't totally independent. If one number is divisible by 3, it changes the odds for the others.
The authors calculated the "average behavior" of these linked coin flips. Surprisingly, this simple probability calculation gave them the exact same constant that the complex X-ray (Tauberian theorem) gave them!

The Big Takeaway

This paper is a victory for mathematical intuition.

Prediction: They guessed the answer using probability and complex analysis.
Proof: They used heavy-duty math (Tauberian theorems) to prove the answer is at least that big.
Verification: They showed that the "random chance" explanation matches the "rigorous math" explanation.

In simple terms: They successfully mapped the "shape" of a very complex number pattern, proving that even in the chaotic world of integers, there is a hidden, predictable rhythm when you look at three numbers standing together. They didn't just guess the rhythm; they proved the music exists, even if they haven't written down every single note of the song yet.

1. Problem Statement

The paper investigates the asymptotic behavior of triple convolution sums involving generalized divisor functions. Specifically, the authors study the sum:
$T(d_k, d_l, d_m; x, h) := \sum_{n \le x} d_k(n+h) d_l(n) d_m(n-h)$
where:

$d_k(n)$ is the generalized divisor function (the number of ways to write $n$ as a product of $k$ positive integers).
$h$ is a fixed integer shift such that $h \le x^{1-\epsilon}$ for any $\epsilon > 0$ .
The case $k=l=m=2$ corresponds to the standard divisor function $d(n)$ .

The authors also extend this investigation to sums over prime numbers, motivated by the Titchmarsh divisor problem:
$T'(d_k, d_m; x, h) := \sum_{p \le x} d_k(p+h) d_m(p-h)$
(Note: Since $d_l(p) = l$ for a prime $p$ , the middle term in the triple sum over primes reduces to a constant, effectively making it a shifted double convolution sum over primes).

The Core Challenge: While asymptotic formulas for double convolution sums (e.g., $\sum d(n)d(n+h)$ ) are well-established (Ingham, Conrey-Gonek), the asymptotic formula for the triple convolution sum remains an open problem for $k, l, m \ge 2$ . Previous work by Browning and others provided "average" results or lower bounds, but a precise asymptotic with a recovered constant from first principles was lacking.

2. Methodology

The authors employ a multi-faceted approach combining analytic number theory, probabilistic heuristics, and Tauberian theorems.

A. Probabilistic Heuristics

The authors first derive a conjectured asymptotic formula using probabilistic arguments (following Tao and Ng-Thom). They model the divisor functions as random variables indexed by prime factors. By calculating the expected values of products of these local random variables, they predict the main term of the asymptotic expansion, including a specific arithmetic constant $\nabla_{k,l,m}(h)$ .

B. Multiple Dirichlet Series and Tauberian Theorems

To rigorously prove a lower bound of the correct order, the authors utilize the theory of Multiple Dirichlet Series (MDS).

Decomposition: They decompose the convolution sum using the identity $d_k(n) = \sum_{u|n} d_{k-1}(u)$ . This transforms the problem into counting solutions to a system of congruences involving divisors.
Generating Functions: They construct a multivariable Dirichlet series $F(s_1, s_2, s_3)$ associated with the sum.
Euler Product Analysis: They analyze the Euler product of $F$ , splitting it into factors for primes dividing $2h$ and those that do not. This allows them to relate $F$ to powers of the Riemann zeta function, $\zeta(1+s_i)$ .
Tauberian Theorems: They apply the de la Bretèche Tauberian theorems (specifically Theorems 2.1 and 2.2 from de la Bretèche's work). These theorems allow the extraction of the asymptotic behavior of partial sums from the analytic properties (poles and growth) of the Dirichlet series. Crucially, this method recovers the exact constant predicted by the probabilistic heuristic.

C. Sums Over Primes (Bombieri-Vinogradov)

For the sum over primes, the authors adapt the method to handle the indicator function of primes.

They express the sum using the condition that $n$ is prime.
They utilize the Bombieri-Vinogradov theorem (and its weighted variant by Nair and Tenebaum) to handle the distribution of primes in arithmetic progressions with large moduli.
This allows them to replace the sum over primes with a sum over integers weighted by the logarithmic integral, controlling the error terms effectively.

3. Key Contributions and Results

Contribution 1: Conjectured Asymptotic and Constant Recovery

The paper establishes a precise conjecture for the asymptotic behavior of the triple sum:
$T(d_k, d_l, d_m; x, h) \sim \frac{\nabla_{k,l,m}(h)}{(k-1)!(l-1)!(m-1)!} x (\log x)^{k+l+m-3}$
The constant $\nabla_{k,l,m}(h)$ is explicitly defined as a product over primes involving local densities and arithmetic conditions.

Contribution 2: Rigorous Lower Bound (Theorem 1.1)

The authors prove that the lower bound matches the predicted order of magnitude and, significantly, recovers the exact constant predicted by the probabilistic model and Dirichlet series theory:
$T(d_k, d_l, d_m; x, h) \ge \frac{\nabla_{k,l,m}(h)}{3^{k+l+m-3}} \frac{x (\log x)^{k+l+m-3}}{(k-1)!(l-1)!(m-1)!} + O(x(\log x)^{k+l+m-4})$
Note: The factor $3^{k+l+m-3}$ arises from the specific truncation of the summation range in the lower bound proof, but the constant $\nabla$ itself is the "correct" one.

Contribution 3: Lower Bound for Prime Sums (Theorem 1.2)

They derive an explicit lower bound for the shifted convolution sum over primes:
$T'(d_k, d_m; x, h) \ge \frac{\nabla'_{k,m}(h)}{4^{k+m-2}} \frac{x (\log x)^{k+m-3}}{(k-1)!(m-1)!} + O(x(\log x)^{k+m-4})$
This result extends the Titchmarsh divisor problem to higher-order divisor functions and triple correlations.

Contribution 4: Verification of Constants

The paper provides a detailed verification (Section 5) showing that the constant $\nabla_{k,l,m}(h)$ derived via the complex analytic method (Tauberian theorems) is identical to the constant derived via the probabilistic heuristic (expectations of random variables). This unifies two distinct approaches to the problem.

4. Significance

Advancement of Convolution Theory: This work pushes the boundary of convolution sums from the well-understood double case to the challenging triple case for generalized divisor functions.
Methodological Unification: It demonstrates the power of combining probabilistic heuristics (for prediction) with multiple Dirichlet series and Tauberian theorems (for rigorous proof). The ability to recover the exact constant from the analytic method validates the probabilistic model.
Extension to Primes: By successfully applying these techniques to sums over primes using the Bombieri-Vinogradov theorem, the authors open new avenues for studying correlations of arithmetic functions in the context of prime numbers, a notoriously difficult area in analytic number theory.
Explicit Constants: Unlike many asymptotic results that only determine the order of magnitude ( $O(\cdot)$ ), this paper provides explicit formulas for the leading constants, which are crucial for numerical verification and deeper theoretical understanding.

In summary, Misra and Saha provide a comprehensive framework for analyzing triple convolution sums, offering the first rigorous lower bounds with the correct leading constants for these sums, both over integers and over primes.