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A Kubilius model for sieve-theoretic sequences

This paper establishes a qualitatively optimal bound on the total variation distance for the Kubilius model applied to sequences with a positive level of distribution, thereby recovering and simplifying recent results on shifted primes while providing a streamlined proof of Tenenbaum's optimal bound for the classical case.

Original authors: Ofir Gorodetsky

Published 2026-08-17

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Ofir Gorodetsky

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 trying to guess the secret recipe of a giant, chaotic soup. In the world of numbers, this soup is the collection of all whole numbers, and the "ingredients" are prime numbers (like 2, 3, 5, 7, 11). Every whole number is made by multiplying these primes together in different amounts. For example, 12 is $2 \times 2 \times 3$ . The big question mathematicians have asked for decades is: if you pick a random number, how predictable are its ingredients? Does it have a lot of 2s? A few 3s? Or is it a total mystery?

To solve this, mathematicians use a clever trick called a "model." Instead of trying to track the messy, real ingredients of every single number, they build a pretend version where the ingredients are chosen completely at random, like rolling dice. If the real world behaves just like the dice game, the model is a success. This is the "Kubilius model," named after the mathematician who first proposed it. It's a bit like predicting the weather: if your computer model says there's a 50% chance of rain, and it actually rains half the time, your model is good. But if the real world has hidden patterns the dice don't know about, the model fails. The goal is to measure exactly how far off the real world is from the random dice game.

This paper, written by Ofir Gorodetsky, is about sharpening the ruler we use to measure that distance. The author isn't just checking if the model works; he's finding the absolute best possible way to prove how well it works, especially when we look at very large numbers. He takes a powerful tool called "sieve theory" (which is like a kitchen strainer that separates big ingredients from small ones) and combines it with some clever math tricks to get a much tighter, more accurate measurement than anyone had before. The result is a proof that shows the random dice model is incredibly close to reality, almost as close as mathematically possible.

The Story of the Dice and the Soup

Let's dive into the main discovery. Imagine you have a giant jar of numbers, and you pick one at random. You want to know the "recipe" of that number: how many times does the prime number 2 divide it? How many times does 3 divide it? And so on. In the real world, these counts are linked together in complicated ways. But in the Kubilius model, we pretend they are independent, like rolling a separate die for each prime number.

The paper asks: How different is the real recipe from the fake, random one? Mathematicians measure this difference using something called "total variation distance." Think of it as a "mismatch score." If the score is zero, the real world and the random model are identical twins. If the score is high, they are strangers.

Gorodetsky's main finding is a new, super-precise formula for this mismatch score. He proves that for a wide range of numbers, the difference between the real world and the random model is incredibly small. In fact, he shows that the error drops off so fast that it's almost negligible once you get to large enough numbers. It's like saying, "If you roll a billion dice, the pattern you get is almost indistinguishable from the pattern of a billion real numbers."

Why the Old Rules Needed an Upgrade

Before this paper, mathematicians had a few ways to measure this mismatch. One famous method, developed by a mathematician named Elliott, was good but a bit clunky. It was like using a ruler made of rubber; it gave you a general idea, but it stretched a bit, making the measurements less precise. Another method, by Tenenbaum, was very sharp but required using extremely complex tools (complex analysis) that were hard to apply to different types of numbers.

Gorodetsky's paper bridges the gap. He takes the flexible, easy-to-use "rubber ruler" approach of Elliott and tightens it up until it's as sharp as Tenenbaum's laser, but without needing the heavy machinery. He does this by borrowing a clever trick from another mathematician, Kevin Ford, who worked on "shifted primes" (numbers like $p+1$ where $p$ is a prime). Ford had found a way to handle the messy parts of the problem by ignoring the "bad" outcomes and focusing only on the "good" ones. Gorodetsky realized this trick could be applied to the general problem of all numbers, not just shifted primes.

The "Sieve" and the "Bad" Numbers

To understand how he did it, imagine you are trying to count the number of people in a stadium who are wearing red hats. The "sieve" is a method to filter out everyone who isn't wearing a red hat. In math, sieves help us count numbers with specific properties.

The paper uses a "fundamental lemma of sieve theory," which is a powerful rule that tells us how well a sieve works. Gorodetsky uses this rule to separate the numbers into two groups:

The "Good" Group: Numbers that behave exactly like the random dice model.
The "Bad" Group: Numbers that are weird outliers and don't fit the pattern.

The genius of the paper is in how it handles the "Bad" Group. Instead of trying to count them perfectly (which is hard), the author shows that the "Bad" Group is so small that it doesn't matter much. He proves that the error caused by these outliers is tiny, much smaller than previous estimates allowed.

The Result: A Qualitatively Optimal Bound

The paper concludes with a result that the author believes is "qualitatively optimal." This is a fancy way of saying, "We can't really do much better than this without changing the rules of the game." The formula he derives shows that the mismatch score drops off at a rate that is essentially the best possible.

For example, if you look at numbers up to a certain size $x$ , and you only care about prime factors up to a size $y$ , the error depends on a ratio called $u$ (which is roughly $\log x / \log y$ ). The paper proves that the error is roughly $u^{-u}$ . This means that as $u$ gets bigger (meaning you are looking at larger numbers or a wider range of primes), the error shrinks incredibly fast—faster than you might expect.

The paper also recovers a recent result by Ford regarding "shifted primes" (numbers like $p+a$ ) but with a simpler proof. This is like solving a puzzle that someone else just solved, but finding a path that is shorter and easier to walk. It confirms that the random model works perfectly for these shifted primes too, with a very high degree of certainty.

What This Means for the Future

The paper doesn't just say "we found a better number." It provides a new, robust toolkit for mathematicians. Because the proof is built on flexible "sieve" arguments, it can be adapted to many different situations. Whether you are studying the factors of random numbers, the factors of polynomials, or even the cycle structures of random permutations (which is like shuffling a deck of cards), this new bound gives a clearer picture of how random these structures really are.

The author is careful to note that while the bound is "optimal" in its general shape, there are still tiny factors (like $x^{o(1)}$ ) that might be tweaked in the future. But for all practical purposes, the gap between the real world and the random model has been measured with the highest precision currently possible.

In short, Gorodetsky has taken a messy, complicated problem in number theory and cleaned it up. He showed that the universe of numbers, for all its apparent chaos, follows the rules of a simple dice game with astonishing accuracy. And he did it by finding a better way to count the exceptions, proving that the exceptions are far fewer and less dangerous than we thought.

Problem Statement
The paper addresses the problem of quantifying the accuracy of the Kubilius model in probabilistic number theory. Specifically, it seeks to bound the total variation distance ( $d_{TV}$ ) between the distribution of prime factor exponents of a random integer (or a sequence of integers with a positive level of distribution) and a corresponding tuple of independent random variables.

In the classical setting, let $N_x$ be a random integer chosen uniformly from $[1, x]$ . The vector of exponents $(\nu_p(N_x))_{p \le y}$ , where $\nu_p(n)$ is the exponent of prime $p$ in $n$ , is modeled by a tuple of independent geometric random variables $(G_{p})_{p \le y}$ with $P(G_p = k) = p^{-k}(1 - 1/p)$ . The central question is how small the distance $d_{TV}((\nu_p(N_x))_{p \le y}, (G_p)_{p \le y})$ is as a function of $u = \log x / \log y$ .

Previous work by Kubilius, Elliott, and Tenenbaum established bounds for this distance. Kubilius showed the distance decays exponentially in $u$ . Elliott generalized these results to sequences defined by polynomials and shifted primes, obtaining bounds of the form $u^{-u/8}$ or $u^{-u/75}$ . Tenenbaum later used complex analysis to prove an optimal bound of the form $u^{-u}$ (up to small factors). Ford recently established a sharp analogue for shifted primes $P_{x,a} + a$ , improving the exponent in the error term significantly.

Methodology
The author, Ofir Gorodetsky, employs a sieve-theoretic approach that refines the methods of Kubilius and Elliott while incorporating specific technical innovations from Ford's work on shifted primes.

Total Variation Distance Formula: Instead of using the standard definition involving the supremum over subsets (which requires bounding the probability of "bad" sets for both the arithmetic and probabilistic models), the paper utilizes the identity:
$d_{TV}(X, Y) = \sum_{\omega \in \Omega} \max\{P(X=\omega) - P(Y=\omega), 0\}$
This formulation allows the author to bound the contribution of "bad" outcomes using only the probability under the arithmetic model ( $X$ ), avoiding the need for difficult bounds on the probabilistic model ( $Y$ ) regarding smoothness or other arithmetic constraints.
Fundamental Lemma of Sieve Theory: The proof relies heavily on the Brun–Hooley sieve (a variant of the pure sieve). The author uses a specific form of the fundamental lemma (Proposition 3.1) to estimate the probability that a random variable is coprime to a product of primes. This lemma provides an asymptotic formula with an explicit error term involving a parameter $s = \log D / \log z$ .
Parameter Optimization: A key technical step involves the careful selection of parameters $T$ and $S$ (where $S \ge Ty$ ). The error terms are split into two components:
- $D_1$ : An error term depending only on the multiplicative function $g$ and the parameters, representing the tail of the distribution.
- $D_2$ : An error term encoding the "level of distribution" of the input sequence $A$ , involving sums of $|P(d|A) - g(d)|$ .
  The author optimizes the choice of $T$ to balance these terms, utilizing Rankin's trick and estimates for the density of $y$ -smooth numbers to bound $D_1$ .

Key Contributions and Results
The main result is Theorem 2.3, a sharpening of Elliott's Theorem 1.1. It provides a bound for the total variation distance between $(\nu_p(A))_{p \in P}$ and independent random variables $(G_p)_{p \in P}$ defined by a multiplicative function $g$ . The bound takes the form:
$d_{TV} \ll D_1 + D_2$
where $D_1$ is roughly of size $u_T^{-u_T}$ (with $u_T = \log T / \log y$ ) and $D_2$ is a sum over divisors up to $S$ measuring the deviation of the arithmetic sequence from the model.

The paper derives several corollaries:

Corollary 2.1: Recovers the classical case for $N_x$ . It yields a bound of $u^{-u(1-\epsilon)} + x^{-1+\epsilon}$ , which is a slightly weaker version of Tenenbaum's optimal bound (1.9) but is derived via a simpler, sieve-theoretic proof. The paper notes that the $\epsilon$ can be replaced by an explicit quantity tending to 0.
Corollary 2.2 & 2.5: Applies the result to shifted primes and polynomial sequences. Specifically, it recovers Ford's result on shifted primes (bound 1.11) with a simplified proof. It improves the exponent in the error term for polynomial sequences $h(N_x)$ and $h(P_x)$ compared to previous results by Uˇzdavinis and Barban (improving exponents like $1/50$ and $1/75$ to $1-\epsilon$ and $1/2-\epsilon$ respectively).

Significance and Claims
The paper claims to provide a result that is "qualitatively optimal." By refining the sieve-theoretic arguments and adopting the clever parameter choices from Ford's work, the author achieves bounds that match the strength of Tenenbaum's complex-analytic results in the classical case and Ford's results for shifted primes, but with a unified and simplified proof structure.

The significance lies in:

Simplification: It offers a simpler proof for Tenenbaum's optimal bound in the classical case, avoiding the heavy machinery of complex analysis.
Unification: It provides a single framework that handles classical integers, shifted primes, and polynomial sequences, recovering and improving upon disparate results in the literature.
Optimality: The author asserts that the obtained bounds are qualitatively optimal, matching the known lower bounds for the total variation distance in these settings.

The paper does not propose new experimental applications or future implications beyond the theoretical refinement of these probabilistic models in number theory. The work is presented as a technical improvement on existing sieve-theoretic methods.

The Story of the Dice and the Soup

Why the Old Rules Needed an Upgrade

The "Sieve" and the "Bad" Numbers

The Result: A Qualitatively Optimal Bound

What This Means for the Future

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