Primes have level of distribution four sevenths with respect to well-factorable weights.
5 events · 1 assessment · 3 decisions
Structured and assessed
First pass. Identified the claim as the Bombieri–Friedlander–Iwaniec theorem (Acta Math. 1986, Theorem 10): level of distribution 4/7 for well-factorable weights. Reworded the canonical form from the popular-press "adjusted level" framing to the precision the literature uses, keeping direction. Decomposition kept light for a settled theorem: one named argument (the BFI dispersion-method proof) with a single requires subclaim, the Deshouillers–Iwaniec Kloosterman sum bounds (match_claim: novel; minted at importance 0.15 as a deferred stub). Considered minting Maynard's 3/5 triply-well-factorable extension (match_claim: novel) but it is lateral, not a dependency, and there is no tool to mint a standalone lateral claim, so it is described in prose instead. Importance set to 0.15 / contestation 0.03 (settled, undisputed). Corrected the Quanta instance: filled speaker/publication/date from the byline, replaced the elided verbatim passage with the contiguous sentence so the quote check can find it. Recorded two new affirming instances (Maynard 2020 arXiv abstract; Lichtman arXiv 2109.02851). Assessed verified at confidence 0.95, credence 0.99, marginal yield 0.05. Source map written as immaterial. No dependents exist, so no notification. Tool note: provenance_read_source failed on the arXiv PDF with a UTF8 byte error; used the abstract page instead.
Assessed Verified
verdict confidence 0.95 · credence 0.99
This is the theorem of Bombieri, Friedlander and Iwaniec published in Acta Mathematica in 1986 ("Primes in arithmetic progressions to large moduli"). The unconditional Bombieri–Vinogradov theorem controls the error in the prime number theorem for arithmetic progressions on average over moduli up to about the square root of x, that is, level of distribution one half. Bombieri, Friedlander and Iwaniec showed that if the moduli are weighted by a well-factorable sequence, one that can be split as a convolution at any chosen scale, the average error remains small for moduli up to x^{4/7-ε}. Since the upper-bound weights of the linear sieve admit a well-factorable variant, this "adjusted" level of 4/7 feeds directly into sieve applications, which is why it is often described as raising the level of distribution of the primes for sieve purposes. The proof combines Linnik's dispersion method with the Deshouillers–Iwaniec bounds for sums of Kloosterman sums, themselves a standard result drawn from the spectral theory of automorphic forms. The theorem has been refereed, independently expounded and used continuously for four decades, and no objection to it has appeared. Later work has strengthened it for narrower classes of weights: Maynard reached moduli up to x^{3/5-ε} for triply well-factorable weights in 2020, and Lichtman reached x^{10/17} for linear sieve weights with strong factorization properties. These extensions supersede the 4/7 level in their own settings without disturbing the original result, which remains the reference point they are measured against.
Updated claim instance
Instance 21047a9c-e79d-4b5e-af46-ead69490ec12 (https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/): was stance=affirms, confidence=0.9, speaker=Erica Klarreich; set verbatimText="During the late 1980s, three researchers — Enrico Bombieri, a Fields medalist at the Institute for Advanced Study in Princeton, John Friedlander of the University of Toronto, and Henryk Iwaniec of Rutgers University — had developed a way to tweak the definition of the level of distribution to bring the value of this adjusted parameter up to 4/7 .". The recorded passage was an elided splice with ellipses and did not match the stored text mechanically. Replaced with the full contiguous sentence as it appears in the article, so the quotation can be verified against the source.
Updated claim instance
Instance 21047a9c-e79d-4b5e-af46-ead69490ec12 (https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/): was stance=affirms, confidence=0.8; set confidence=0.9, speaker="Erica Klarreich", publication="Quanta Magazine", sourceDate="2013-05-19". Read the source whole: the article is by Erica Klarreich for Quanta Magazine, dated May 19, 2013, and states in its own voice that Bombieri, Friedlander and Iwaniec brought the adjusted level of distribution up to 4/7. Stance affirms is correct; metadata was missing and is filled from the byline and dateline. Confidence raised slightly: the passage is a direct, unhedged assertion of the claim, though in popular rather than technical wording.
Claim entered the graph