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Complete Asymptotic Expansion of the Additive Mertens Sum $S_k(x)$

This paper establishes the complete asymptotic expansion of the additive Mertens sum $S_k(x)$ , deriving its leading order behavior of $x^{k-1}/\log^k x$ and providing explicit formulas for the expansion coefficients in terms of multiple logarithmic integrals, with fully explicit results for the first few terms for general $k$ and a complete closed-form sequence for the case $k=2$ .

Original authors: Daoyi Peng, Hao Liu

Published 2026-07-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daoyi Peng, Hao Liu

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 have a giant bag of prime numbers—the building blocks of math like 2, 3, 5, 7, and so on. Now, imagine you pick $k$ of these primes at random (where $k$ is at least 2) and add them up. The paper asks a very specific question: If you take the reciprocal of that sum (1 divided by the sum) and add up every possible combination of these primes up to a huge number $x$ , what does the total look like?

This is called the additive Mertens sum, denoted as $S_k(x)$ .

The Big Surprise: It's Not What You Expected

For a long time, mathematicians were obsessed with a multiplicative version of this problem. In that version, instead of adding the primes, you multiply them (like $p_1 \times p_2 \times \dots$ ). That version grows very slowly, like a whisper that gets louder only as you double-click a button over and over (mathematically, it grows like $(\log \log x)^k$ ).

But this paper flips the script. The authors, Daoyi Peng and Hao Liu, discovered that when you add the primes instead of multiplying them, the sum explodes much faster. It doesn't whisper; it shouts. The leading term of this sum grows like $\frac{x^{k-1}}{(\log x)^k}$ .

To visualize this: If the multiplicative version is a snail crawling up a wall, the additive version is a rocket blasting off, but with a very specific, predictable fuel consumption rate.

The Master Recipe: A Complete Asymptotic Expansion

The paper doesn't just tell you the rocket's speed; it writes out the complete recipe for the entire flight path. They proved that as $x$ gets infinitely large, you can describe $S_k(x)$ with a precise formula:

$S_k(x) \approx \frac{x^{k-1}}{(\log x)^k} \times \left( \text{Constant}_0 + \frac{\text{Constant}_1}{\log x} + \frac{\text{Constant}_2}{(\log x)^2} + \dots \right)$

Think of this like a high-end telescope. The first term ( $\text{Constant}_0$ ) gives you a blurry picture of the stars. The next term ( $\text{Constant}_1$ ) sharpens the focus. The next one ( $\text{Constant}_2$ ) makes the image crystal clear. The authors proved you can keep adding these "focus knobs" forever to get a better and better approximation.

The Secret Ingredients: The Coefficients

The real magic lies in figuring out what those "Constants" actually are. The authors found that these numbers aren't random; they are hidden inside complex, multi-dimensional integrals (imagine measuring the volume of a shape that exists in $k$ dimensions, where the shape is defined by logarithms).

They managed to crack the code for the first few ingredients:

The First Ingredient ( $E_{k,0}$ ): They found a clean, closed-form formula for this. For example, if you have 2 primes ( $k=2$ ), this number is $2 \log 2$ .
The Second Ingredient ( $E_{k,1}$ ): This one is a bit more complex, involving famous math constants like $\pi^2$ and special functions called polylogarithms ( $\text{Li}_2$ ).
The Third Ingredient ( $E_{k,2}$ ): Here, things get tricky. The authors could write down a perfect formula for the "diagonal" part of this ingredient (where the variables behave nicely), but the "cross" part (where variables mix in messy ways) becomes a nightmare for $k \ge 5$ .

Crucially, the paper explicitly rules out a simple, unified formula for the third ingredient when $k$ is large (5 or more). While they can calculate it for small cases like $k=2, 3, 4$ , the math gets so tangled with strange polylogarithm values that they admit a single, neat "closed-form" expression for all $k$ likely doesn't exist in the way we hope. They don't just say "it's hard"; they show that the structure fundamentally changes, introducing new types of numbers that don't play nice with the old ones.

The Special Case: $k=2$

When you only pick two primes ( $k=2$ ), the authors found a "Golden Ticket." They discovered that the entire sequence of coefficients ( $E_{2,0}, E_{2,1}, E_{2,2}, \dots$ ) follows a single, beautiful pattern involving the Dirichlet eta function and the Riemann zeta function. It's like finding that the entire infinite series of focus knobs for a specific telescope can be described by one single, elegant equation.

How Sure Are They?

The authors are extremely confident in their main result. They didn't just guess; they proved it using rigorous tools like the Prime Number Theorem and multivariate Taylor expansions.

The Proof: They mathematically demonstrated that the error in their formula shrinks faster than any power of $\log x$ as $x$ gets huge.
The Numbers: To back up their theory, they ran massive computer simulations.
- For $k=2$ , they tested numbers up to $5,000,000$. The approximation was off by only 0.2% when using just the first five terms of their formula.
- For $k=3$ , they tested up to $10,000,000$. The three-term formula was off by about 1.3%.
- They also noticed something fascinating: if you stop the formula too early for small numbers, it gets worse. This is the nature of an "asymptotic" series—it's like a map that gets more accurate the further you travel, but if you try to use the "far away" part of the map to navigate your backyard, you might get lost.

The Bottom Line

This paper solves a long-standing mystery about how sums of prime reciprocals behave when you add them up. It replaces a vague guess with a precise, infinite recipe. While they cracked the code for the first few steps of the recipe (and the whole recipe for the simplest case of two primes), they also drew a clear line in the sand: for more complex cases with many primes, the "cross-term" ingredients are too messy to be written down in a single, simple formula. The math is there, it's proven, and it's been tested, but the final piece of the puzzle for large $k$ remains a beautiful, unsolved knot.

Technical Summary: Complete Asymptotic Expansion of the Additive Mertens Sum $S_k(x)$

Problem Statement
The paper addresses the asymptotic behavior of the additive Mertens sum, defined for an integer $k \ge 2$ and $x > 2$ as:
$S_k(x) = \sum_{p_1 \le x} \cdots \sum_{p_k \le x} \frac{1}{p_1 + \cdots + p_k},$
where $p_i$ are primes. While the multiplicative analogue (generalized Mertens sum $\sum 1/(p_1 \cdots p_k)$ ) is well-studied with a leading term of order $(\log \log x)^k$ , the additive sum exhibits a fundamentally different growth rate. Elementary bounds derived from the AM-GM inequality and the Prime Number Theorem (PNT) suggest the leading term is of order $x^{k-1}/\log^k x$ . The primary objective of this work is to establish the complete asymptotic expansion of $S_k(x)$ , determining the exact coefficients of the series and providing closed-form expressions for the initial terms.

Methodology
The authors employ a combination of analytic number theory and real-variable analysis techniques:

Stieltjes Integral Representation: The sum is reformulated as a $k$ -fold Stieltjes integral involving the prime counting function $\pi(u)$ .
Prime Number Theorem Decomposition: The differential $d\pi(u)$ is split into the main term $du/\log u$ and a remainder term $dR(u)$, where $R(u) = \pi(u) - \text{Li}(u)$ . The error terms arising from the remainder are shown to be negligible (decaying faster than any power of $\log x$ ) using integration by parts and bounds on $R(u)$ .
Variable Rescaling and Taylor Expansion: The main term integral is rescaled via $u_j = x t_j$ . The domain is partitioned into a "main region" (where $t_j$ are bounded away from 0) and a "boundary region." In the main region, the kernel $\prod (L + \log t_j)^{-1}$ (where $L = \log x$ ) is expanded using a multivariate Taylor series involving complete homogeneous symmetric polynomials $h_n$ .
Laplace Transform Techniques: The resulting coefficient integrals, defined over the unit cube $(0,1]^k$ , are transformed into single-variable integrals using the identity $1/(\sum t_j) = \int_0^\infty e^{-x \sum t_j} dx$ . This reduces the problem to analyzing generating functions satisfying specific higher-order differential equations.
Finite Difference Extraction: The coefficients of the expansion are extracted using forward difference operators $\Delta^{k-1}$ applied to the generating functions, leveraging the property that differences of polynomials of degree less than $k-1$ vanish.

Key Contributions and Results

Complete Asymptotic Expansion (Theorem 1): The paper proves that for any fixed $N \ge 0$ :
$S_k(x) = \frac{x^{k-1}}{\log^k x} \sum_{n=0}^N \frac{E_{k,n}}{\log^n x} + O\left(\frac{x^{k-1}}{\log^{k+N+1} x}\right).$
The coefficients $E_{k,n}$ are given by absolutely convergent multiple logarithmic integrals:
$E_{k,n} = (-1)^n \int_{(0,1]^k} \frac{h_n(\log t_1, \dots, \log t_k)}{t_1 + \cdots + t_k} dt.$
Closed Forms for Initial Coefficients (Theorem 2):
- Leading Term ( $E_{k,0}$ ): A closed form is derived involving a finite sum of logarithms weighted by binomial coefficients and powers.
- Second Order ( $E_{k,1}$ ): The coefficient is expressed in a basis involving $\pi^2$ , $\log m$ , and the dilogarithm $\text{Li}_2(-1/m)$ .
- Third Order ( $E_{k,2}$ ): The coefficient is decomposed into a "diagonal" part ( $k A_k$ ) and a "cross" part ( $\binom{k}{2} B_k$ ). The diagonal part admits a closed form for all $k$ involving $\zeta(3)$ , $\pi^2$ , $\text{Li}_3(-1/m)$ , and logarithmic sums. The cross part is fully explicit for $k \le 4$ but becomes structurally complex for $k \ge 5$ , involving polylogarithms with denominators $\ge 5$ that do not appear to reduce to the standard basis.
Explicit Expansions for $k=2$ and $k=3$ :
- For $k=2$ , the paper provides a closed formula for the entire sequence $\{E_{2,n}\}_{n \ge 0}$ . The coefficients are explicit $\mathbb{Q}$ -linear combinations of $\log 2$ and Riemann zeta values $\zeta(j)$ , derived using the Dirichlet eta function.
- For $k=3$ , the first three terms of the expansion are fully explicit, combining the results for $E_{3,0}, E_{3,1}, E_{3,2}$ .
Numerical Verification: The authors perform numerical computations for $S_2(x)$ and $S_3(x)$ up to $x = 10^7$ . The results confirm that the relative error decreases as $x \to \infty$ and that adding successive terms improves accuracy, consistent with the asymptotic nature of the series. The data also illustrates the phenomenon of optimal truncation for fixed $x$ .

Significance and Claims
The paper claims to provide the first study of the additive analogue of the Mertens sum in the literature. Its primary significance lies in:

Establishing the correct order of magnitude ( $x^{k-1}/\log^k x$ ) and the complete asymptotic structure of the sum, contrasting it with the multiplicative case.
Reducing the computation of asymptotic coefficients to the evaluation of specific multiple integrals and providing explicit closed forms for the leading and sub-leading terms.
Identifying a structural divergence in the coefficients for $k \ge 5$ , where the "cross" terms introduce polylogarithmic values that likely resist reduction to a unified closed form, highlighting a boundary in current analytic capabilities for such sums.

The work relies strictly on the real-variable form of the PNT and does not invoke deep conjectures (like the Riemann Hypothesis) for the main asymptotic results, though the error terms depend on the classical PNT error bound. The paper concludes that while the diagonal parts of higher-order coefficients remain tractable via generating functions, the cross parts present a fundamental obstacle to a unified closed form for general $k$ .