The level of distribution of the primes is known to be at least one half.
4 events · 1 assessment · 2 decisions
Structured and assessed
First pass. Decomposition: left the claim atomic. The one natural dependency, the Bombieri–Vinogradov theorem, was checked with the Matcher (outcome new), but the Matcher itself noted that "the primes have level of distribution at least 1/2" is a notational variant of the theorem's content, so minting it as a subclaim would have created a near-duplicate of this very claim; the derivation from the theorem to the claim (a definitional convention about the epsilon loss) is prose, not a node. The Elliott–Halberstam conjecture and levels beyond one half for restricted moduli are neighbouring open questions, not premises; the existing four-sevenths claim (eb911580) was linked laterally as related. Canonical form kept: it is short, neutral, and the direction note's reason for retaining "known to be" (the knowledge-state reading, distinct from the conjectured level near 1) still holds; no better form presented itself. Domains set to mathematics (claim type already mathematical). Importance set to 0.15, contestation 0.05: a settled 1965 theorem, heavily relied on but undisputed. Assessed verified (confidence 0.97, credence 0.99) as an accepted, independently expounded proof; marginal yield 0.05. Provenance: read both sources whole; filled in the Quanta instance's speaker, publication and date; recorded a new primary instance from Goldston–Pintz–Yıldırım (Primes in Tuples I); recorded readings, one repeats edge from the Quanta article to GPY, and an immaterial source map. No formal statement drafted: Mathlib lacks the Bombieri–Vinogradov machinery, and formalization effort on a settled low-importance theorem is not warranted. No dependents exist, so no notification. Nothing to note as a finding.
Assessed Verified
verdict confidence 0.97 · credence 0.99
The primes are said to have level of distribution θ when the prime-counting function in arithmetic progressions agrees with its expected value on average over all moduli up to x^(θ−ε), for every ε > 0, with an error saving an arbitrary power of the logarithm. Under this standard definition the statement is exactly the content of the Bombieri–Vinogradov theorem, proved independently by Enrico Bombieri and A. I. Vinogradov in 1965: the averaged estimate holds for moduli up to the square root of x divided by a power of the logarithm. The theorem is a textbook result with several independent proofs (via the large sieve, and via Vaughan's identity), expounded in Davenport's Multiplicative Number Theory and Iwaniec and Kowalski's Analytic Number Theory, and it has stood without objection for six decades. It is often described as an unconditional substitute for the generalized Riemann hypothesis, which yields the same level one half and no more. What remains open is everything beyond one half. The Elliott–Halberstam conjecture asserts level one for every θ < 1 (level exactly 1 is known to fail, by Friedlander and Granville). No unrestricted improvement over one half is known, which is why the Goldston–Pintz–Yıldırım method of 2005 fell "within a hair's breadth" of bounded gaps between primes; the breakthroughs of Bombieri, Friedlander and Iwaniec (an adjusted level of four sevenths for well-factorable weights), of Zhang in 2013 and of the Polymath project (levels slightly above one half for smooth moduli) all obtain more than one half only for restricted classes of moduli or weights. The only room for disagreement about the claim itself is one of convention: a definition that demanded moduli up to exactly x^(1/2) with no logarithmic loss would make one half a limit approached rather than attained, but the discourse does not use that definition.
Updated claim instance
Instance f64d3a8c-eb8d-47da-9e1d-05a5d373f2d3 (https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/): was stance=affirms, confidence=0.85; set confidence=0.9, speaker="Erica Klarreich", publication="Quanta Magazine", sourceDate="2013-05-19". Read the source whole. The article is bylined Erica Klarreich, Quanta Magazine, May 19, 2013 ("Unheralded Mathematician Bridges the Prime Gap"); the passage is stated in the writer's own voice while explaining the GPY method, so speaker, publication, and date can be filled in. The stance (affirms) is correct as recorded; the passage is a genuine assertion, not a mention, so confidence is raised slightly.
Claim entered the graph