The level of distribution of the primes is known to be at least one half.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
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.
Full reasoning: the evidence and decisions behind this verdict
The claim is a knowledge-state restatement of the Bombieri–Vinogradov theorem, so its truth reduces to two questions: whether the theorem is established, and whether it delivers level one half under the definition of level of distribution the discourse uses.
On the first: Bombieri (Mathematika 12, 1965) and Vinogradov (Izv. Akad. Nauk SSSR 29, 1965) proved that for any A > 0 there is B such that the sum over q ≤ x^(1/2) (log x)^(−B) of the maximum over coprime residue classes of |ψ(x; q, a) − x/φ(q)| is O(x (log x)^(−A)). The proof has been simplified and re-derived several times (Gallagher's large-sieve treatment, Vaughan's identity), appears in standard graduate texts, and is used as a black box throughout sieve theory and the small-gaps literature. There is no unresolved objection in the literature; this meets the mathematics standard for an accepted proof.
On the second: Goldston, Pintz and Yıldırım (Primes in Tuples I, arxiv.org/html/math.NT/0508185) define the primes to have level of distribution θ if the estimate holds for every A and every ε > 0 with Q = N^(θ−ε), and state that under this definition the Bombieri–Vinogradov theorem gives level 1/2 and Elliott–Halberstam conjectures level 1. Soundararajan's Bulletin AMS survey of the GPY work uses the same convention and states the same conclusion. Since x^(1/2)(log x)^(−B) exceeds x^(1/2−ε) for every fixed ε and large x, the theorem yields level one half exactly under this definition, and "at least one half" holds under any supremum-style definition as well. The only reading under which the claim would be strained is a non-standard one requiring Q = x^(1/2) with no logarithmic loss; no source in the discourse uses it.
Instances: both recorded sources affirm the claim. The Goldston–Pintz–Yıldırım paper is a primary technical statement with the definition attached; the Quanta article of 19 May 2013 by Erica Klarreich (www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/) repeats it as background while explaining the GPY method and Zhang's result, and adds no evidence of its own. No source denies the claim, and the surrounding literature (GPY, Zhang, Polymath8, Maynard) uniformly treats one half as the known unconditional level and anything beyond it, for unrestricted moduli, as open.
Adversarial check: the strongest challenge would be that "at least one half" suggests a strict inequality is known; it does not, and no source reads it so. A second would be that later work (Zhang, Polymath8, Bombieri–Friedlander–Iwaniec) has moved the known level above one half; those results apply only to smooth moduli or well-factorable weights and so do not contradict the claim, and they are recorded as a related claim rather than as evidence against this one. Neither challenge moves the verdict. What would change the conclusion: nothing short of an error being found in the Bombieri–Vinogradov theorem itself, which after sixty years of use and multiple independent proofs is not a live possibility.
Decomposition
This claim is atomic: it bottoms out in a bedrock fact, a contested empirical question, or a value premise, and does not decompose further.
Provenance
Where this claim has been said, linked to its canonical form.
The level of distribution is known to be at least ½. This is exactly the right value to prove the GPY result, but it falls just short of proving that there are always pairs of primes with a bounded gap.
Discussing the parameter used in the GPY sieve method.
Asserted without evidence of the source's own. A popular account of Zhang's bounded-gaps theorem. It states the known level one half as accepted background without naming the Bombieri–Vinogradov theorem it comes from, and offers no argument of its own; the statement restates the framing of the Goldston–Pintz–Yıldırım paper it is describing.
Elliott and Halberstam [5] conjectured that the primes have level of distribution 1. According to the Bombieri-Vinogradov theorem, the primes are known to have level of distribution 1/2.
Introduction defining the level of distribution (the estimate of Bombieri–Vinogradov type holding for Q = N^(θ−ε) for every ε>0) and stating the known level 1/2 against the Elliott–Halberstam conjecture of level 1, before Theorem 1 which assumes level θ>1/2.
The source's own evidence bears what it asserts. The Goldston–Pintz–Yıldırım paper gives the precise definition in use (the averaged estimate holds for moduli up to N to the power theta minus epsilon, for every epsilon) and states the Bombieri–Vinogradov theorem explicitly with its logarithmic saving, from which level one half follows directly under that definition. This is the primary technical statement of the claim in the small-gaps literature. Worth reading closely: It states the definition of level of distribution under which the value one half is exactly what Bombieri–Vinogradov gives, which is the one point of convention a reader might question.
How these sources relate
- https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/ restates https://arxiv.org/html/math.NT/0508185, faithfully. The article's passage on the level of distribution is an exposition of the GPY paper and quotes it directly in the next paragraph; the GPY introduction states the same proposition ("the primes are known to have level of distribution 1/2") with attribution to Bombieri–Vinogradov. The article adds no evidence of its own and states the proposition at the same strength, so the crossing is faithful. The article does not cite GPY for this specific sentence, hence moderate confidence.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.