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On additive convolution sum of arithmetic functions and related questions

This paper extends Ingham's classical results on additive convolution sums by deriving asymptotic formulas with error terms for partial sums of the divisor function and generalizing these findings to arithmetic functions possessing absolutely convergent Ramanujan expansions.

Original authors: Bikram Misra, Biswajyoti Saha, Anubhav Sharma

Published 2026-06-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bikram Misra, Biswajyoti Saha, Anubhav Sharma

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 involving two numbers that add up to a specific target, let's call it $N$ .

In the world of mathematics, there is a famous puzzle called the Goldbach Conjecture. It asks: "Can every even number be written as the sum of two prime numbers?" (For example, $14 = 3 + 11$ or $14 = 7 + 7$ ).

To investigate this, mathematicians use a tool called a convolution sum. Think of this as a giant tally counter. You walk through every number $n$ from 1 up to $N$ , and you check: "Does $n$ have a special property? Does $N-n$ also have that property?" If both do, you add a point to your score.

The Old Detective Work (Ingham's Discovery)

Decades ago, a mathematician named Ingham studied a specific version of this game. He looked at the divisor function, denoted as $d(n)$ .

What is $d(n)$ ? It's simply the count of how many numbers can divide $n$ evenly. For example, $d(6) = 4$ because 1, 2, 3, and 6 divide 6.
The Game: Ingham counted how many pairs $(n, N-n)$ exist where both numbers have divisors. He found a neat formula that predicts the total score for the entire range from 1 to $N$ as $N$ gets huge.

However, Ingham only looked at the whole picture. He didn't stop to ask: "What if we only count the pairs where the first number is small? What if we stop counting halfway through?"

The New Discovery (The Paper's Contribution)

The authors of this paper, Misra, Saha, and Sharma, decided to zoom in. They wanted to know what happens if we only count the pairs where the first number, $n$ , is less than some smaller number $M$ (where $M$ is less than $N$ ).

Think of it like this:

The Full Sum: Counting every way to fill a swimming pool with water.
The Sub-Sum: Counting only the water in the first few feet of the pool.

The authors found that if you choose your cutoff point $M$ carefully, you can predict the score for this "partial pool" with a very precise formula.

The "Goldilocks" Zone for $M$

The paper reveals that the size of $M$ matters a lot. It's not just "any number smaller than $N$ ."

Too small: If $M$ is tiny (like a speck of dust compared to $N$ ), the math gets messy and their current tools can't give a clean answer.
Just right: If $M$ is a significant chunk of $N$ (but not the whole thing), they found a beautiful pattern. The score for the partial sum is roughly proportional to how much of the pool you filled, but with a twist: the "logarithmic" part of the formula changes based on exactly where you stopped.

They proved that if you stop at $M$ , the formula looks like this:
$\text{Score} \approx \text{Constant} \times M \times \text{Logarithm}(\text{Something related to } M \text{ and } N)$

This is a big deal because it extends Ingham's old work. It tells us that the "density" of these divisor pairs isn't uniform; it shifts slightly depending on how far you go into the number line.

The "Ramanujan" Connection

The paper doesn't just stop at counting divisors. They also looked at more complex arithmetic functions (mathematical rules that assign a number to every integer).

They used a special tool invented by the legendary mathematician Srinivasa Ramanujan called Ramanujan Expansions.

The Analogy: Imagine you have a complex song (a complicated arithmetic function). Ramanujan showed that you can break this song down into a series of simple, pure musical notes (called Ramanujan sums).
The Application: The authors used this "musical breakdown" to analyze the partial sums of these complex functions. They showed that if the "notes" in the song get quiet fast enough (mathematically, if the coefficients decay quickly), you can predict the partial sum with high accuracy.

Why Does This Matter?

The paper connects this math to two famous problems:

Twin Primes: Pairs of primes that are close together (like 3 and 5, or 11 and 13).
Goldbach's Conjecture: The sum of two primes.

The authors explain that if we want to prove Goldbach's conjecture, we might need to understand not just the total number of ways to write $N$ as a sum, but specifically the ways where the smaller prime is below a certain limit. Their new formulas give mathematicians a sharper tool to look at these specific "sub-sums."

Summary in Plain English

This paper is like upgrading a map.

Before: We had a map showing the total terrain from point A to point B.
Now: The authors have drawn a new map that tells us exactly what the terrain looks like if we stop at point C (where C is somewhere between A and B).
The Catch: The map is only perfectly accurate if point C isn't too close to A. If it's too close, the terrain is too rugged for their current tools.
The Result: They provided a new, precise formula for this "partial journey," using clever tricks involving the history of number theory (Ingham and Ramanujan) to solve a problem that had been largely ignored until now.

They didn't solve the Goldbach conjecture itself, but they handed the detectives a much better magnifying glass to look at the clues.

Technical Summary: On Additive Convolution Sums of Arithmetic Functions and Related Questions

Problem Statement
The paper addresses the asymptotic behavior of additive convolution sums of arithmetic functions, specifically focusing on the sub-sum $\sum_{n < M} f(n)g(N-n)$ where $M \le N$ . While the shifted convolution sum $\sum_{n \le N} d(n)d(n+h)$ has been extensively studied since Ingham's seminal work, the additive convolution sum $\sum_{n < N} d(n)d(N-n)$ and its sub-sums have received less attention. The authors aim to extend Ingham's classical results to derive asymptotic formulas for these sub-sums with explicit error terms, particularly for the divisor function $d(n)$ , and subsequently generalize these findings to arithmetic functions admitting absolutely convergent Ramanujan expansions.

Methodology
The authors employ a combination of classical analytic number theory techniques and modern estimates regarding divisor functions in arithmetic progressions.

Extension of Ingham's Method (Theorem 1):
For the divisor function $d(n)$ , the authors adapt Ingham's geometric approach. They interpret the sum $\sum_{n \le M} d(n)d(N-n)$ as counting 4-tuples $(l, r, m, s)$ satisfying $lr + ms = N$ with $ms \le M$ .
- Optimization of Bounds: Unlike Ingham's use of the AM-GM inequality to bound the product $n(N-n)$ by $N^2/4$ , the authors utilize the specific constraint $n \le M$ to derive a tighter bound $X = \sqrt{M(N-M)}$ .
- Set Decomposition: The set of solutions is decomposed into symmetric sets $P$ and $Q$ and an intersection $R$ . The cardinality is estimated using properties of the divisor function and the distribution of solutions to linear Diophantine equations.
- Range Analysis: The proof carefully distinguishes between different ranges of $M$ relative to $N$ , specifically analyzing the error terms when $M$ is close to $N/2$ versus when $M$ is smaller.
Small $M$ Regime (Theorem 2):
For the case where $M = N^\alpha$ with $3/4 < \alpha < 1$ , the geometric method is insufficient. The authors utilize the identity $d(n) = \sum_{t|n} 1$ to rewrite the sum as a sum over residue classes.
- They apply a result by Pongsriiam and Vaughan (extending Fouvry and Iwaniec) regarding the average of the divisor function in arithmetic progressions: $\sum_{n \le M, n \equiv N \pmod t} d(n)$ .
- This allows the derivation of an asymptotic formula involving Ramanujan sums $c_r(N)$ and specific series $S_i(N)$ .
General Arithmetic Functions (Theorem 3):
The authors generalize the results to functions $f$ and $g$ with absolutely convergent Ramanujan expansions ( $f(n) = \sum a_f(r)c_r(n)$ ).
- Orthogonality: They utilize an orthogonality lemma for Ramanujan sums: $\sum_{n < M} c_r(n)c_s(N-n) \approx \delta_{r,s} M c_r(N)$ .
- Truncation and Error Control: The sums are split into a main term (where indices $r, s$ are small) and an error term (where indices are large). The error terms are controlled using the decay conditions $|a_f(r)|, |a_g(r)| \ll r^{-(1+\delta)}$ and the previously established bounds for the divisor function convolution.

Key Contributions and Results

Theorem 1 (Divisor Function Sub-sums): The paper establishes an asymptotic formula for $\sum_{n \le M} d(n)d(N-n)$ for $M \le N/2$ .
- The main term is $\frac{6}{\pi^2} M \sigma_{-1}(N) \log^2 X$ , where $X = \sqrt{M(N-M)}$ .
- The error term depends on the relationship between $M$ and $N$ . Specifically, if $N/(\sigma_{-1}(N) \log N)^2 = o(M)$ , the formula holds.
- The authors provide refined error bounds for different ranges of $M$ , including cases where $M$ is close to $N/2$ and cases where $M$ is smaller but still satisfies the growth condition.
- Corollary 1 extends these results to the range $N/2 \le M \le N$ by utilizing the symmetry of the full sum.
Theorem 2 (Small $M$ ): For $M = N^\alpha$ with $3/4 < \alpha < 1$ , the authors derive a more complex asymptotic formula involving series $S_0(N), S_1(N), S_2(N)$ defined via Ramanujan sums. This fills a gap where the geometric method of Theorem 1 fails.
Theorem 3 (General Functions): For arithmetic functions with absolutely convergent Ramanujan expansions satisfying polynomial decay conditions on their coefficients, the paper proves:
$\sum_{n < M} f(n)g(N-n) = M \sum_{r \ge 1} a_f(r)a_g(r)c_r(N) + \text{Error Term}$
The error term is $O(M^{1-\delta}(\log M)^{4-2\delta})$ for $\delta < 1$ , $O(\log^3 M)$ for $\delta = 1$ , and $O(1)$ for $\delta > 1$ .
Corollary 2: As an application, the authors derive asymptotic formulas for sums involving normalized divisor functions $\sigma_\alpha(n)/n^\alpha$ , improving upon previous results in [6] by providing explicit error terms and extending the range of parameters.

Significance and Claims
The paper claims to extend the classical results of Ingham from full sums to sub-sums, providing a more granular understanding of how the additive convolution behaves as the summation limit $M$ varies.

Motivation: The study is motivated by problems in additive number theory, such as the Goldbach conjecture, where understanding the distribution of prime summands (or prime powers) up to a certain bound $M \le N/2$ is crucial. The authors note that while the full sum counts all representations, the sub-sum behavior is relevant for determining the least possible prime summand.
Methodological Contribution: The work highlights the necessity of carefully choosing the range of $M$ to obtain valid asymptotic formulas with manageable error terms. It demonstrates that for small $M$ , the behavior is governed by the distribution of divisors in arithmetic progressions, while for larger $M$ , geometric constraints on the product $n(N-n)$ dominate.
Limitations: The authors modestly note that their method for Theorem 1 cannot treat very small values of $M$ (specifically $M = N^\alpha$ for $\alpha < 3/4$ ), and the proof for Theorem 2 is restricted to $3/4 < \alpha < 1$ . They also note that the implied constants in the error terms for general functions may depend on $N$ , though this dependence can be mitigated under specific growth conditions on $M$ .

The paper does not claim to solve the Goldbach conjecture or the twin prime conjecture but posits that the derived asymptotic formulas for convolution sub-sums provide necessary tools and insights for such investigations.