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A weaker but simpler sieve inequality

This paper introduces a simplified sieve inequality based on a cancellation property of sieve weights, which is particularly effective for analyzing the distribution of primes and almost-primes within very short intervals.

Original authors: John B. Friedlander

Published 2026-07-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: John B. Friedlander

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: Counting Primes with a Sieve

Imagine you are trying to find specific rare items (like prime numbers) hidden inside a huge pile of rocks. To do this, mathematicians use a "sieve." Think of a sieve not just as a kitchen tool, but as a complex set of rules or a filter. You pour the rocks through the filter, and the filter is designed to let the "bad" numbers (composite numbers) fall through while keeping the "good" numbers (primes) on top.

However, the filter isn't perfect. Sometimes it accidentally throws away good rocks, or it keeps some bad ones. To fix this, mathematicians assign "weights" to the rocks. These weights are like little tags that say, "This rock is probably good," or "This rock is probably bad." The goal is to arrange these tags so that when you add them all up, the bad tags cancel each other out, leaving you with a clear count of the good rocks.

The Problem: The Old Filter Was Too Heavy

For decades, mathematicians (including the author, Friedlander, and his colleague Henryk Iwaniec) had been using a very powerful, but very heavy and complicated, set of tags (weights) to count primes.

They had a formula to measure how well their filter worked. It involved a big sum of squares (imagine squaring the weight of every rock and adding them up).

The Old Method: They proved that this big sum was small enough to be useful. But the proof was like trying to lift a heavy boulder with a complex machine. It worked, but it was messy, required very specific conditions, and was hard to tweak.
The Flaw: It turned out there was a tiny crack in the machine's design (a small flaw in the proof) that a colleague, K. Matomäki, pointed out.

The Discovery: A Letter from the "Grandmaster"

The author was cleaning out his office and found an old letter from Atle Selberg, a legendary mathematician, dated 1981. In the letter, Selberg had answered a question the author had asked him 40+ years ago.

Selberg said, "Yes, you can get a bound, but you don't need the heavy machine. You just need a slightly different, simpler calculation."

The author realized that while Selberg's method proved a slightly weaker result (it didn't measure the entire heavy boulder, just a smaller, lighter piece of it), it was much simpler to prove and didn't have the flaws of the old method.

The "Magic Trick": The Identity

The core of the paper is a mathematical "identity" (a fancy way of saying an equation that is always true).

The Old Way (The Heavy Sum): The author had been calculating a sum called $W$ . This was like trying to weigh every single rock in the pile individually, then squaring those weights, then adding them all up. It was a massive, confusing calculation.
The New Way (The Lighter Sum): Selberg showed that if you calculate a slightly different sum called $U$ (which involves a specific function called Euler's totient function, $\phi$ ), the math becomes incredibly neat.

The Analogy:
Imagine you want to know the total weight of a stack of books.

Method A (The Old Way): You take every book, weigh it, square the number, and add them up. Then you realize you made a mistake in how you stacked them, so you have to re-calculate everything.
Method B (The New Way): Selberg discovered a trick. He showed that if you arrange the books in a specific way and look at the spaces between them, the total weight is actually just the sum of the weights of the books divided by the number of pages they have. It's a shortcut.

The paper proves that this "shortcut" sum ( $U$ ) is actually equal to a very clean, simple product of numbers. This makes it easy to prove that the sum is small (which means the sieve is working well).

Why Does This Matter?

The author gives two main reasons for writing this paper:

It's All You Need: Even though the new method measures a "smaller" sum than the old one, it turns out that for the specific problem of finding primes in very short intervals (looking at a tiny slice of the number line, like finding primes between 1,000,000 and 1,000,100), the "lighter" sum is actually all the mathematician needs. You don't need to lift the whole boulder; you just need to lift the handle.
It's More Natural: The new method feels more "honest." The math flows better. The old method required forcing the numbers to behave in a certain way, while the new method lets the numbers behave naturally.

The "Lambda" Weights

The paper also discusses two types of "tags" (weights) used in sieves:

Beta-sieve: These tags are simple (like $+1$ or $-1$). The new method works perfectly here.
Selberg's $\Lambda^2$ sieve: These tags are more complex (they are products of other numbers). The paper shows that even with these complex tags, the "shortcut" method still works, provided you choose the tags correctly.

Summary

John Friedlander is essentially saying: "We spent 45 years building a complex, heavy machine to count primes. We found a small crack in it. Then, I found an old letter from a master mathematician who told me, 'You don't need the whole machine. There is a simpler, lighter tool that does the job just as well for the specific task you are doing.' This paper is me finally building that simpler tool and showing everyone how it works."

The result is a weaker inequality (it proves a slightly smaller result) but a simpler proof (it's easier to understand and less prone to errors), which is exactly what is needed to study how primes are distributed in short intervals.

Technical Summary: "A Weaker But Simpler Sieve Inequality"

Problem Statement
The paper addresses the estimation of specific sums involving sieve weights, a central task in analytic number theory concerning the distribution of primes and almost-primes. The author focuses on two related sums defined over a set of primes $P$ and a product $P(z) = \prod_{p \in P, p < z} p$ . Let $\lambda_d$ be a sequence of real numbers supported on divisors $d$ of $P(z)$ satisfying $|\lambda_d| \leq \tau_k(d)$ .

The primary object of study is the sum $W$ :
$W = \sum_{d|P} d \left( \sum_{\substack{m|P \\ m \equiv 0 \pmod d}} \frac{\lambda_m}{m} \right)^2$
Previously, in Friedlander and Iwaniec's work [3], a bound for $W$ was established (Lemma 6.18), showing that for specific sieve weights (beta-sieve), $W \ll \prod_{p|P} (1 - 1/p)$ . However, the proof of this result contained a minor flaw identified by K. Matomäki, and the derivation was noted to be complex.

The paper also considers a "smaller" sum $U$ , which was the subject of a 1981 letter from Atle Selberg to the author:
$U = \sum_{d|P} \phi(d) \left( \sum_{\substack{m|P \\ m \equiv 0 \pmod d}} \frac{\lambda_m}{m} \right)^2$
where $\phi$ is Euler's totient function. While $U$ is technically a smaller sum than $W$ (since $\phi(d) < d$ ), the paper argues that $U$ is the more natural object for certain applications and admits a significantly simpler proof.

Methodology
The core methodological contribution is the derivation and utilization of a specific algebraic identity that transforms the sum $U$ into a form amenable to trivial bounding.

The Fundamental Identity: The paper establishes the identity:
$\sum_{d|P} \phi(d) \left( \sum_{\substack{m|P \\ m \equiv 0 \pmod d}} \frac{\lambda_m}{m} \right)^2 = \prod_{p \in P} \left(1 - \frac{1}{p}\right) \sum_{\delta|P} \frac{\theta_\delta^2}{\phi(\delta)}$
where $\theta_m = \sum_{d|m} \lambda_d$ . This identity is derived by interchanging the order of summation and utilizing properties of the greatest common divisor and the Euler totient function.
Comparison with $W$ : The author contrasts this with the identity required for the sum $W$ , which involves a more complicated structure (referenced as identity (4.2) in [3] and (57) in [6]). The identity for $U$ is described as "neat" and "easy to show," whereas the treatment for $W$ is "rather more complicated."
Bounding Strategy: To bound the right-hand side of the identity for $U$ , the author employs a trivial bound on the term $\theta_\delta^2$ . Specifically, using the inequality $\theta_\delta^2 \leq \tau(\delta)|\theta_\delta|$ , the sum is reduced to a form involving divisor functions. The author notes that for sieve weights (specifically beta-sieve and Selberg $\Lambda^2$ -sieve), the coefficients behave as if restricted to almost-primes, keeping the divisor function $\tau(\delta)$ small enough to ensure the sum remains bounded.
Generalization: The paper extends these identities to a more general setting involving multiplicative functions $f$ and $f'$ (where $f'(p) = f(p) - 1$ ), leading to a generalized identity (3.1) that encompasses the specific case of sieving intervals.

Key Results

Proof of Selberg's Conjecture: The paper provides a proof of the bound for the sum $U$ that was conjectured by the author in [1] and confirmed by Selberg in his 1981 letter. The result is:
$\sum_{d|P} \phi(d) \left( \sum_{\substack{m|P \\ m \equiv 0 \pmod d}} \frac{\lambda_m}{m} \right)^2 \ll \prod_{p|P} \left(1 - \frac{1}{p}\right)$
This holds for both beta-sieve weights and Selberg $\Lambda^2$ -sieve weights.
Simplification of Proof: The derivation of the bound for $U$ is presented as "much more quickly and cleanly" than the proof for the stronger bound on $W$ .
Correction of Previous Work: The paper acknowledges a flaw in the proof of the bound for $W$ in [3] and notes that a corrected proof is provided in the appendix (joint with Iwaniec).
Application to Short Intervals: The author states that the bound for $U$ is "all that is actually needed" for applications regarding the distribution of primes and almost-primes in short intervals (specifically intervals of length $g(x)\log x$ where $g(x) \to \infty$ ). In previous estimations (e.g., equation 6.120 in [3]), the count of integers coprime to $d$ was bounded by $H$ rather than the more precise $H\phi(d)/d$ ; the sum $U$ naturally accommodates this more precise counting.

Significance and Claims
The paper claims modest significance, framing itself as a "weaker but simpler" alternative to previous results.

Naturalness: The author argues that $U$ is the "more natural" sum compared to $W$ , justified by the simplicity of the underlying identity (2.3).
Sufficiency: The primary claim is that while the bound on $W$ is stronger, the bound on $U$ is sufficient for the specific application of estimating primes and almost-primes in short intervals.
Historical Context: The paper serves to highlight a result from a 1981 letter by Atle Selberg that had been overlooked or forgotten, providing a modern, streamlined proof of the inequality Selberg confirmed.
No New Applications: The paper does not propose new applications or future implications. It strictly limits its scope to the theoretical simplification of existing sieve inequalities and their relevance to established problems in short intervals, noting that recent developments by K. Matomäki have further utilized these types of results.

In summary, the paper offers a streamlined, algebraically elegant proof for a specific sieve inequality ( $U$ ) that, while technically weaker than a previously known result ( $W$ ), is sufficient for key applications in prime number theory and avoids the complexities and flaws of the earlier proofs.