Primes have level of distribution four sevenths with respect to well-factorable weights.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
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.
Full reasoning: the evidence and decisions behind this verdict
The claim is a settled theorem, so the assessment rests on the published proof and its standing in the literature rather than on a fresh check of every step. The primary source is Bombieri, Friedlander and Iwaniec, Acta Math. 156 (1986), 203–251, whose Theorem 10 states: for a well-factorable sequence λ of level Q ≤ x^{4/7-ε}, the weighted sum over q of the error terms ψ(x; q, a) − x/φ(q) is O(x (log x)^{-A}) for any A, with the implied constant depending on a, A and ε. The proof treats the error terms by the dispersion method, uses the factorability of λ to arrange the bilinear forms at convenient scales, and bounds the resulting off-diagonal sums of Kloosterman sums with the Deshouillers–Iwaniec estimates of 1982; the sole named result the argument turns on is therefore the Deshouillers–Iwaniec bounds for sums of Kloosterman sums, which are themselves undisputed.
The three recorded instances all affirm the claim and are independent voices, though none is the original paper. Maynard's 2020 paper (arxiv.org/abs/2006.07088) restates the BFI theorem as its Theorem A and treats it as established, then extends it to x^{3/5-ε} for triply well-factorable weights. Lichtman's paper (arxiv.org/pdf/2109.02851) states the BFI level x^{4/7} as the benchmark its own x^{10/17} result for linear sieve weights surpasses. The Quanta Magazine article of 2013 (www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/) states the result in popular terms as an adjusted level brought up to 4/7. No source denying the claim was found, and none would be expected: the theorem has been reused in Chen's theorem refinements, in the analysis of Zhang's bounded gaps work, and in the recent Maynard and Lichtman extensions, all of which take it as given.
The claim is precise as stated once "well-factorable" is understood in the Iwaniec sense (a sequence of level Q that, for every factorisation Q = Q1 Q2, can be written as a convolution of bounded sequences supported on [1, Q1] and [1, Q2]). The extensions to 3/5 and 10/17 are for narrower weight classes and are distinct claims; they do not bear on whether 4/7 holds for well-factorable weights in general, and 4/7 remains the best level for that general class in the literature read for this pass. Verified status rests on an accepted, refereed, independently expounded proof that has stood for four decades, not on a machine-checked formalization; no formal statement exists for this claim and none was attempted at this importance. What would change the verdict is a published error in the 1986 argument or in the Deshouillers–Iwaniec bounds as used there, neither of which has appeared.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Bombieri, Friedlander and Iwaniec apply Linnik's dispersion method to the error terms of primes in progressions weighted by a well-factorable sequence, and because such a sequence factors at any chosen scale the resulting bilinear forms can be arranged so that their off-diagonal terms are sums of Kloosterman sums; given the Deshouillers–Iwaniec bounds for sums of Kloosterman sums, those terms are small for moduli up to x^{4/7-ε}, which yields the theorem.
The inference goes through: the dispersion method reduces the weighted error terms to bilinear forms whose off-diagonal contribution is a sum of Kloosterman sums, and the factorability of the weights lets the scales be chosen so that the Kloosterman bounds win up to exponent 4/7. The argument lives on the Deshouillers–Iwaniec bounds for sums of Kloosterman sums, a standard refereed result in continuous use since 1982, so nothing in its premises is in doubt. The proof was refereed in Acta Mathematica in 1986 and has been independently expounded and reused since, most recently as the starting point of Maynard's and Lichtman's extensions.
Provenance
Where this claim has been said, linked to its canonical form.
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 .
On late-1980s work adjusting the level of distribution.
Asserted without evidence of the source's own. A popular science account that states the Bombieri, Friedlander and Iwaniec result correctly in outline, without stating the technical condition (well-factorable weights) or citing the 1986 paper. It offers no evidence of its own and is a faithful restatement of the published theorem. The quoted passage was not found in the stored copy of this source.
This extends well-known work of Bombieri, Friedlander and Iwaniec, who handled moduli of size at most $x^{4/7-\epsilon}$.
Abstract of a paper extending the Bombieri–Friedlander–Iwaniec well-factorable estimate from moduli up to x^{4/7-ε} to x^{3/5-ε} for triply well-factorable weights; states the BFI 4/7 result as established.
This surpasses the level of distribution x^{4/7} with the linear sieve weights from well-known work of Bombieri, Friedlander, and Iwaniec
Abstract of a paper equidistributing primes to moduli up to x^{10/17} for linear sieve weights with strong factorization; states the BFI 4/7 level as the established prior benchmark it surpasses.
Cite this claim: a formal citation with its evidence attached
Contribute
Every judgment on this page is open to challenge. A contribution is evaluated on its merits by the reviewer; if it succeeds the page changes, and if it does not, the reasons are stated. Either way the exchange becomes part of the claim’s public record.
The attention this claim received was paid for by a funded mandate. Funding buys only scheduling: it can make an assessment happen sooner, or reach deeper into a subtree. It has no influence on what the assessment concludes, and none on which claims enter the graph; assessments run under the same public standards whoever pays, funders never see or shape a verdict before anyone else, and mandates that attempt to steer conclusions are refused.
Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.