A weaker but simpler sieve inequality

John B. Friedlander

A weaker but simpler sieve inequality

T0 review · 0 major / 4 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read A simpler identity for sieve weights yields the same short-interval cancellation with less work.

desk verdict Clean expository note that publicizes Selberg's 1981 identity for the weaker sum U, shows it already covers the short-interval applications, and fixes the Opera lemma gap Matomäki flagged. read the letter →

arxiv 2607.05707 v1 pith:AMTWLOOI submitted 2026-07-07 math.NT

John B. Friedlander This is my paper · ORCID

classification math.NT MSC 11N0511N35

keywords sieveweightsbeta-sieveSelbergLambda-squaredshortintervalsprimesalmost-primescancellationEulertotient

topics The Twin Prime Conjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Sieve weights that detect primes or almost-primes can cancel strongly when summed in a certain quadratic form. Earlier work already proved that a weighted sum W is bounded by the product of (1-1/p) over the sieved primes; that bound controls primes and almost-primes in almost all short intervals. This note shows that a slightly smaller sum U, which replaces the factor d by Euler's totient phi(d), obeys exactly the same bound and is all that the short-interval applications actually need. The proof rests on a clean identity that rewrites U as a product times a sum of squares of the partial sums of the weights; a crude but serviceable estimate then reduces the claim to a standard two-dimensional sieve inequality. The same identity and bound hold for both beta-sieve weights and suitable Selberg Lambda-squared weights, giving a shorter and more natural route to the cancellation that short-interval results require.

What carries the argument

The identity (2.3) that rewrites U as product_p (1-1/p) times sum_delta theta_delta^2 / phi(delta), where theta_m = sum_{d|m} lambda_d; a trivial bound theta^2 <= tau |theta| (or tau_2 theta) then reduces the problem to a standard sieve estimate of dimension 2.

What would settle it

Compute the sum of theta_delta^2 / phi(delta) explicitly for beta-sieve weights with beta < 8 and check whether it remains bounded independently of the sifting range; if it grows, the final O(1) claim fails for those smaller beta.

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Extended reading notes

Core claim

For beta-sieve weights (or suitable Lambda-squared weights) the sum U = sum_d phi(d) (sum_{m: d|m} lambda_m/m)^2 is << product_p (1-1/p). This is the same order of magnitude previously obtained for the larger sum W, and is precisely the quantity needed for the short-interval applications.

Load-bearing premise

The argument needs the sieve level parameter beta to be at least 8 so that a standard two-dimensional sieve inequality can absorb the extra divisor growth introduced by the crude bound on the squares of the weight sums.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

John's note is short and useful. The punchline is that the weaker sum U (with φ(d)) already gives the same O(∏(1-1/p)) bound that the stronger W did, via the elementary identity (2.3) that Selberg wrote him in 1981. That identity reduces everything to bounding ∑ θ_δ^{2}/φ(δ), which is cleaner than the corresponding treatment of W. He also records a corrected proof of Opera Lemma 6.18 that tracks the coprimality conditions Matomäki pointed out, so the statement of the lemma is unchanged but the derivation is now solid.

What is actually new is modest: publicizing the identity, showing that the short-interval applications only ever needed U (the earlier papers had bounded the count of h by H rather than Hφ(d)/d), and writing down the general multiplicative form (3.1). The bound itself was already known in stronger form, and the identity is not original to this paper. The math is elementary and transparent; the appendix carefully rewrites the earlier argument. Citations are appropriate (his own earlier papers, Opera, Matomäki, Selberg).

The soft spot is exactly the one the reader flagged: the crude majorant θ^{2} ≤ τ|θ| (or τ_{2}θ for Λ^{2}) forces the large β ≥ 8 and s ≥ β+1 before one can invoke the dimension-2 sieve estimates. John himself notes that this may have cost a chance at a smaller β. That is a genuine limitation of the method, not a flaw in the argument; the final bound still holds under the stated hypotheses and is known to be enough for the applications.

This is for people who work with combinatorial or Selberg sieves and short intervals. It is not a research breakthrough, but it is a clean, honest clarification that will save others from reinventing the identity or relying on the flawed Opera write-up. I would send it to a serious referee; the note is short, the correction is real, and the exposition is careful. Worth reading if you care about these weights; not essential if you don't.

Referee Report

0 major / 4 minor

Summary. The paper proves that for beta-sieve weights (or suitable Selberg Lambda-squared weights) the sum U = sum_{d|P} phi(d) (sum_{m: d|m} lambda_m/m)^2 satisfies U << product_{p|P} (1-1/p). An elementary identity (2.3) reduces U to a sum of theta_delta^2/phi(delta), which is then bounded by the same dimension-kappa=2 sieve estimates used for the larger sum W in Opera de Cribro. The bound is sufficient for the short-interval applications that motivate the work; an appendix supplies a corrected proof of the original Opera Lemma 6.18 (addressing a coprimality gap noted by Matomaki) and related identities for general density functions are recorded.

Significance. The result supplies a cleaner, more natural route to the estimate actually needed for upper bounds on primes and lower bounds on almost-primes in almost all intervals of length g(x) log x (g->infty arbitrarily slowly). The identity (2.3) is transparent and self-contained; the appendix correction of Lemma 6.18 is a useful service to the literature. The historical framing via Selberg's 1981 letter adds context without affecting the mathematics. The argument relies only on standard beta-sieve machinery already in wide use, so the contribution is incremental but cleanly executed and immediately applicable.

minor comments (4)

[Appendix] Appendix, line after (4.3): 'Propostion 6.7' is misspelled; correct to 'Proposition'.
[Appendix] Appendix, paragraph containing (4.3): the hyphenation 'dimen- sion kappa=2' should be repaired for readability.
[Section 2] Section 2, after (2.5): the text says the proof of (2.2) 'can be taken to run along the very same lines as does that for (1.3) in Section 4'; a one-sentence reminder that the same conditions beta >= 8 and s >= beta+1 are in force would make the dependence fully explicit without forcing the reader to the appendix.
[References] References: the arXiv identifier of the present paper appears as 2607.05707; if this is a placeholder it should be updated on final submission.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; elementary identity (2.3) is self-contained and the bound follows standard sieve estimates without definitional reduction.

full rationale

The paper's central contribution is the elementary identity (2.3) obtained by direct interchange of summation and complementary divisors, reducing the bound for U to an estimate of sum theta_delta^2 / phi(delta). That estimate proceeds via the crude majorant (2.4) (or (2.8)) and then the same lines as the corrected proof of the stronger bound for W given in the appendix, which invokes only the classical beta-sieve estimates of dimension kappa=2 under the usual level condition s >= beta+1 (beta >= 8). These estimates are standard results of sieve theory (appearing in the authors' prior book and used by many subsequent authors); they do not presuppose the target inequality for U, nor is there any definitional loop, fitted parameter renamed as a prediction, uniqueness theorem imported from the authors, or ansatz smuggled via citation. The appendix itself supplies an independent correction of a minor gap previously noted by Matomaki. Self-citations to Opera de Cribro and earlier papers of the author are present but supply established machinery rather than forcing the conclusion by construction. The derivation is therefore free of the circular patterns listed.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The argument rests on standard multiplicative-function identities, the classical beta-sieve and Lambda-squared weight estimates from Opera de Cribro, and the elementary majorant theta^2 <= tau |theta|. No free parameters are fitted; the sieve level beta >= 8 is an inherited technical hypothesis, not a data-driven constant. No new entities are postulated.

assumptions (3)

domain assumption Beta-sieve weights of level D with beta >= 8 satisfy the dimension-kappa=2 estimates of Opera Proposition 6.7 (upper and lower bounds for sum lambda_c g(c)).
Invoked at the end of Section 2 and again in the appendix to convert the weighted sum of theta into a product of local factors.
domain assumption For beta-sieve weights one has |lambda_d| <= 1, hence theta_m = sum_{d|m} lambda_d satisfies |theta_m| <= tau(m); for Lambda^2 weights |lambda_d| <= 3^{nu(d)} and theta_m <= tau_2(m).
Used to justify the majorants (2.4) and (2.8) that reduce the sum of squares to a linear form controllable by the sieve.
standard math The elementary identity relating sum phi(d) (sum_{d|m} lambda_m/m)^2 to product (1-1/p) times sum theta_delta^2 / phi(delta) holds for any real coefficients lambda.
Proved by double interchange of summation and complementary divisors; appears as (2.3).

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Cite this review

Pith. "Pith review of A weaker but simpler sieve inequality." pith.science (2026). https://pith.science/paper/AMTWLOOI

@misc{pith2026260705707,
  author       = {Pith},
  title        = {Pith review of: A weaker but simpler sieve inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AMTWLOOI}},
  note         = {Machine review of arXiv:2607.05707}
}

read the original abstract

We discuss a cancellation property of sieve weights, one that is applicable to the distribution of primes and almost-primes in very short intervals.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Friedlander, Sifting short intervals, Math

J.B. Friedlander, Sifting short intervals, Math. Proc. Cambridge Philos. Soc. 91(1982), 9–15

work page 1982
[2]

Friedlander, Sifting short intervals II, Math

J.B. Friedlander, Sifting short intervals II, Math. Proc. Cambridge Philos. Soc.92(1982), 381–384

work page 1982
[3]

Friedlander and H

J.B. Friedlander and H. Iwaniec, Opera de Cribro,Colloquium Publications, 57, (xx plus 527 pages), Amer. Math. Soc. (Providence) 2010

work page 2010
[4]

Iwaniec, Rosser’s sieve, Acta Arith.36, (1980) 171–202

H. Iwaniec, Rosser’s sieve, Acta Arith.36, (1980) 171–202

work page 1980
[5]

Matom¨ aki, e-mail to JF and HI, Oct

K. Matom¨ aki, e-mail to JF and HI, Oct. 5, 2020. 10 FRIEDLANDER

work page 2020
[6]

Matom¨ aki, Almost primes in almost all very short intervals

K. Matom¨ aki, Almost primes in almost all very short intervals. J. Lond. Math. Soc. (2) 106 (2022), no. 2, 1061–1097

work page 2022
[7]

Selberg, personal communication, Nov

A. Selberg, personal communication, Nov. 13, 1981

work page 1981
[8]

Selberg, Lectures on sieves, appearing in Collected Papers II, pp 65–247, Springer Verlag (Berlin) 1991

A. Selberg, Lectures on sieves, appearing in Collected Papers II, pp 65–247, Springer Verlag (Berlin) 1991. Department of Mathematics, University of Toronto Toronto, Ontario M5S 2E4, Canada (frdlndr@math.utoronto.ca)

work page 1991

Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.

[1] [1]

Friedlander, Sifting short intervals, Math

J.B. Friedlander, Sifting short intervals, Math. Proc. Cambridge Philos. Soc. 91(1982), 9–15

work page 1982

[2] [2]

Friedlander, Sifting short intervals II, Math

J.B. Friedlander, Sifting short intervals II, Math. Proc. Cambridge Philos. Soc.92(1982), 381–384

work page 1982

[3] [3]

Friedlander and H

J.B. Friedlander and H. Iwaniec, Opera de Cribro,Colloquium Publications, 57, (xx plus 527 pages), Amer. Math. Soc. (Providence) 2010

work page 2010

[4] [4]

Iwaniec, Rosser’s sieve, Acta Arith.36, (1980) 171–202

H. Iwaniec, Rosser’s sieve, Acta Arith.36, (1980) 171–202

work page 1980

[5] [5]

Matom¨ aki, e-mail to JF and HI, Oct

K. Matom¨ aki, e-mail to JF and HI, Oct. 5, 2020. 10 FRIEDLANDER

work page 2020

[6] [6]

Matom¨ aki, Almost primes in almost all very short intervals

K. Matom¨ aki, Almost primes in almost all very short intervals. J. Lond. Math. Soc. (2) 106 (2022), no. 2, 1061–1097

work page 2022

[7] [7]

Selberg, personal communication, Nov

A. Selberg, personal communication, Nov. 13, 1981

work page 1981

[8] [8]

Selberg, Lectures on sieves, appearing in Collected Papers II, pp 65–247, Springer Verlag (Berlin) 1991

A. Selberg, Lectures on sieves, appearing in Collected Papers II, pp 65–247, Springer Verlag (Berlin) 1991. Department of Mathematics, University of Toronto Toronto, Ontario M5S 2E4, Canada (frdlndr@math.utoronto.ca)

work page 1991