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On Maximal Values of Gronwall Numbers for Integers with Given Greatest Prime Factor and Remainder in Modified Mertens Formula

This paper establishes an unconditional, sharp limit relationship as the prime $p$ tends to infinity between the remainder in the modified Mertens asymptotic formula for the sum of prime reciprocals and the maximal values of Gronwall numbers among integers with greatest prime factor $p$ that are divisible by any smaller prime $q < p$ .

Original authors: Gennadiy A. Kalyabin

Published 2026-03-26

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Gennadiy A. Kalyabin

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: A Game of "Perfectly Packed Boxes"

Imagine you have a giant warehouse filled with numbers. Your job is to find the "champion" numbers—those that are the most efficient at packing their own divisors (the numbers that divide into them evenly).

In math, there is a famous rule called the Gronwall Number. Think of this as a "score" for a number. The score is calculated by taking the sum of all its divisors, dividing it by the number itself, and then adjusting for how big the number is.

The Goal: Find the number with the highest possible score.
The Mystery: For a long time, mathematicians knew that these scores get closer and closer to a specific limit (a "ceiling" of about 1.78). But they didn't know exactly how the scores approach that ceiling, or if they ever accidentally jump over it.

The Two Main Characters

The paper focuses on two specific groups of numbers, which we can think of as two different teams in a competition:

The "Primorial" Team (The Strict Team): These are numbers built using every prime number up to a certain point (like 2, 3, 5, 7, 11...). You can't skip any. If you stop at 11, your number must be made of 2, 3, 5, 7, and 11.
The "Greatest Prime" Team (The Flexible Team): These numbers have a specific "biggest" prime factor (say, 11), but they might be missing some smaller primes (like maybe they have 2, 5, and 11, but no 3).

The author, Gennadiy Kalyabin, is trying to figure out the maximum possible score for the "Strict Team" as the numbers get infinitely large.

The Connection to the "Riemann Hypothesis" (The Holy Grail)

There is a famous unsolved puzzle in math called the Riemann Hypothesis. It's like a master key that unlocks many secrets about prime numbers.

The Stakes: In 1983, a mathematician named Robin proved a scary fact: If the "Strict Team" ever produces a number with a score higher than the theoretical limit (1.78...), then the Riemann Hypothesis is false.
The Current Status: We don't know if the Riemann Hypothesis is true or false. So, we don't know if these "super-high scores" exist.

What This Paper Actually Does

Kalyabin's paper is a bit like a detective trying to find the "perfect" number without needing to solve the whole Riemann Hypothesis first. He does this by looking at the remainder (the tiny error) in a famous formula called Mertens' Formula.

Think of Mertens' Formula as a map that predicts how prime numbers are distributed. The map isn't perfect; it has a tiny "error margin" (the remainder).

The Discovery:
Kalyabin found a direct, unbreakable link between:

How high the "Strict Team's" scores can get (the Gronwall numbers).
The tiny error in the prime number map (the Mertens remainder).

He proved that as the numbers get huge, the difference between the highest possible score and the theoretical limit is exactly determined by that tiny error in the map.

The "Unconditional" Breakthrough

Most math papers on this topic say, "If the Riemann Hypothesis is true, then X happens."
Kalyabin says, "It doesn't matter if the Riemann Hypothesis is true or false. This relationship holds true either way."

He calls this an unconditional result. It's like saying, "Whether the sky is blue or green, the grass will always grow." He doesn't need to know the answer to the biggest puzzle in math to prove this specific relationship.

The "Perfectly Packed" Numbers

The paper also describes the structure of the numbers that achieve these maximum scores.

Imagine you are building a tower out of blocks (primes).
To get the highest score, you can't just stack them randomly. You have to follow a very specific, almost rhythmic pattern.
Kalyabin describes exactly how many blocks of each size you need. It turns out the "perfect" numbers are built by taking the first few primes, then skipping a bit, then taking the next few, in a very specific mathematical dance.

The "Unimprovable" Concept

The author introduces a concept called "One-Step G-unimprovable numbers."

Imagine you have a number. If you multiply it by any single prime number, its score goes down.
This means the number is "locally perfect." It's at the top of a hill; if you take one step in any direction (multiply by a prime), you go downhill.
Kalyabin proves that these "local peaks" are the ones that eventually reach the highest possible heights.

The "Limit Relationship" (The Climax)

The main result (Theorem 1) is a formula that connects the dots. It says:

"The gap between the highest score we can find and the theoretical limit is equal to a specific value derived from the error in the prime number map."

He calculates this gap precisely. It involves a square root and a logarithm, but the key takeaway is that he has pinned down the behavior of these numbers with extreme precision, without needing to solve the Riemann Hypothesis.

Summary for the Everyday Reader

The Problem: Mathematicians are hunting for the "best" numbers based on their divisors.
The Risk: If a "better" number exists than expected, it breaks the Riemann Hypothesis (a huge math mystery).
The Solution: Kalyabin found a way to measure exactly how close these numbers get to the limit, using a different formula (Mertens) that we already understand.
The Result: He proved a strict, mathematical rule connecting the "best numbers" to the "error in the prime map." This rule works regardless of whether the Riemann Hypothesis is true or false.

In a metaphor:
Imagine trying to guess the height of a mountain peak. Most people say, "We can only guess if we know the weather (Riemann Hypothesis)." Kalyabin looked at the shadows cast by the trees (the Mertens remainder) and said, "I don't need to know the weather. By measuring the shadows, I can tell you exactly how high the peak is, and I can prove it without ever climbing the mountain."

1. Problem Statement and Context

The paper addresses the asymptotic behavior of Gronwall numbers, defined as $G(N) = \frac{\sigma(N)}{N \log \log N}$ , where $\sigma(N)$ is the sum of divisors of $N$ .

Historical Background: T. Gronwall (1913) established that $\limsup_{N \to \infty} G(N) = e^\gamma$ (where $\gamma$ is the Euler-Mascheroni constant).
The Riemann Hypothesis (RH) Connection: S. Ramanujan and G. Robin proved that the inequality $G(N) < e^\gamma$ holds for all $N > 5040$ if and only if the Riemann Hypothesis is true.
Specific Focus: The author investigates the maximal values of $G(N)$ restricted to integers $N$ whose greatest prime factor is a specific prime $p_k$ . Specifically, the paper focuses on the subset of integers $N$ that are multiples of the primorial $P_k = p_1 \cdots p_k$ (denoted as $\tilde{W}_k$ ).
The Gap: While the limit of $G(N)$ is known, the precise rate of convergence and the relationship between the maximal values in these restricted sets and the remainder term of the Modified Mertens Formula had not been fully characterized unconditionally (i.e., without assuming RH).

2. Methodology

The author employs a combination of analytic number theory, asymptotic analysis, and constructive number theory.

A. Definitions and Sets

$\tilde{W}_k$ : The set of integers $N$ such that $P^+(N) = p_k$ and $N$ is divisible by all primes $p_j \leq p_k$ .
$\tilde{G}_k$ : The maximum value of $G(N)$ for $N \in \tilde{W}_k$ .
Modified Mertens Formula: The paper utilizes the formula $S(x) = \sum_{p \leq x} \frac{\log p}{p-1} = \log \log \theta(x) + \gamma + Q(x)$ , where $Q(x)$ is the remainder term.

B. Structural Characterization of Maximal Integers

The author relies on the concept of "one-step G-unimprovable numbers" (class $U_1$ ). An integer $N$ is in $U_1$ if $G(Np) \leq G(N)$ for any prime $p$ .

Theorem 2 establishes necessary and sufficient conditions for an integer $N \in \tilde{W}_k$ to be a local maximizer of $G(N)$ .
These conditions involve the exponents $\alpha_j$ in the prime factorization $N = \prod p_j^{\alpha_j}$ . Specifically, the exponents must be monotonically non-increasing ( $\alpha_1 \geq \alpha_2 \geq \dots \geq \alpha_k$ ), and the logarithm of $N$ ( $\eta = \log N$ ) must satisfy specific inequalities involving a function $\xi(p, \alpha)$ derived from the derivative of $G(N)$ .
Crucially, for $N \in U_1 \cap \tilde{W}_k$ , the author proves that $\log N \approx \theta_k + \sqrt{2\theta_k}$ , where $\theta_k = \theta(p_k)$ is the first Chebyshev function.

C. Constructive Proofs

Part I (Lower Bound): The author explicitly constructs a sequence of integers $M_k$ with specific exponent patterns (e.g., $p_1^{b_k} \dots p_m^{b_k} \dots p_n^2 \dots p_k^1$ ) to demonstrate that the maximal value $\tilde{G}_k$ can be bounded from below by a specific function involving the remainder $Q(p_k)$ .
Part II (Upper Bound): The author treats the problem as an optimization problem. By relaxing the integer exponents to real variables $t_j$ , the author maximizes the function $\tilde{g}(t) = \log \tilde{G}(t)$ . Using Lagrange multipliers (differentiation), they derive the stationary point and show that the maximum value is strictly bounded by the derived asymptotic formula.

3. Key Contributions and Results

The Main Theorem (Theorem 1)

The paper establishes an unconditional (valid without assuming the Riemann Hypothesis) sharp limit relationship between the deviation of the maximal Gronwall number from $e^\gamma$ and the remainder of the Modified Mertens formula.

The core result is:
$\liminf_{k \to \infty} (\tilde{g}_k - \gamma - Q_k)\sqrt{p_k} \log p_k = -2\sqrt{2}$
Where:

$\tilde{g}_k = \log \tilde{G}_k$ .
$Q_k = Q(p_k)$ is the remainder in the Modified Mertens formula.
$p_k$ is the $k$ -th prime.

This implies two specific bounds for large $k$ :

Lower Bound (Constructive): There exists a sequence of integers $M_k$ such that:
$\log G(M_k) > S(p_k) - \log \log \theta(p_k) - \frac{2\sqrt{2} + \epsilon}{\sqrt{p_k} \log p_k}$
Upper Bound (Universal): For all $N \in \tilde{W}_k$ (specifically those in the set $E$ where maxima exist):
$\log G(N) < S(p_k) - \log \log \theta(p_k) - \frac{2\sqrt{2} - \epsilon}{\sqrt{p_k} \log p_k}$

Structural Insights

The paper characterizes the structure of integers where these maximal values are attained. They are "locally G-maximal" numbers where the exponents of prime factors decrease monotonically, and the total logarithmic size of the number is approximately $\theta_k + \sqrt{2\theta_k}$ .
The author proves that the set of indices $k$ for which such maximal integers exist is infinite.

4. Significance

Unconditional Validity: Unlike many results in this field that depend on the Riemann Hypothesis, these results are proven unconditionally. This provides a solid baseline for understanding the distribution of highly composite numbers.
Refinement of Robin's Inequality: The results provide a precise asymptotic expansion for the "worst-case" scenario of the Robin inequality. The term $-2\sqrt{2}$ quantifies exactly how much the maximal Gronwall number falls short of the theoretical limit $e^\gamma$ relative to the Mertens remainder.
Connection to Chebyshev Functions: The work deeply integrates the properties of the Chebyshev functions ( $\theta(x)$ and $\psi(x)$ ) with the arithmetic properties of divisor sums, offering a new perspective on the interplay between prime distribution and multiplicative number theory.
Future Implications: The author suggests that these findings on the sequence $g_k - \gamma$ (the general case, not just the primorial multiples) could be instrumental in a future proof or disproof of the Ramanujan-Robin inequality (and thus the Riemann Hypothesis).

In summary, Kalyabin's paper provides a rigorous, unconditional asymptotic formula linking the maximal values of Gronwall numbers for integers with a fixed greatest prime factor to the error term in the Modified Mertens formula, identifying a specific constant ( $-2\sqrt{2}$ ) that governs this relationship.