Math 2.0

Number theory Analytic number theory Theorem

Bombieri–Vinogradov theorem

Builds on Chebyshev functionsvon Mangoldt functionEuler's totient functionLarge sieveSiegel–Walfisz theorem

Statement

Let q≥1q \ge 1 and aa be integers, and let y≥2y \ge 2 be real. Let ψ(y;q,a)\psi(y; q, a) be the sum of the von Mangoldt function Λ(n)\Lambda(n) over n≤yn \le y with n≡a(modq)n \equiv a \pmod q. Let φ\varphi be Euler's totient function. For every A>0A > 0 there is B=B(A)>0B = B(A) > 0 such that, for all x≥2x \ge 2 and Q=x1/2(log⁡x)−BQ = x^{1/2} (\log x)^{-B}, ∑q≤Q  max⁡gcd⁡(a,q)=1  max⁡2≤y≤x∣ψ(y;q,a)−yφ(q)∣≤CAx(log⁡x)A,\sum_{q \le Q} \; \max_{\gcd(a, q) = 1} \; \max_{2 \le y \le x} \left| \psi(y; q, a) - \frac{y}{\varphi(q)} \right| \le C_A \frac{x}{(\log x)^{A}}, where CAC_A depends only on AA.

Remarks

The proof combines the large sieve inequality for Dirichlet characters with a decomposition of Λ\Lambda into bilinear sums, such as Vaughan's identity; small moduli are handled by the Siegel–Walfisz theorem. The generalized Riemann hypothesis would give ∣ψ(y;q,a)−y/φ(q)∣≤C y1/2(log⁡y)2\abs{\psi(y; q, a) - y/\varphi(q)} \le C\, y^{1/2} (\log y)^2 for each q≤yq \le y, which yields the same bound for QQ up to about x1/2(log⁡x)−A−2x^{1/2} (\log x)^{-A-2}. So the theorem says that GRH holds "on average over moduli" up to x\sqrt{x}. One says that the primes have level of distribution 1/21/2. The Elliott–Halberstam conjecture states that the same bound holds with Q=xθQ = x^{\theta} for every θ<1\theta < 1.

Going deeper

For a student

The Bombieri–Vinogradov theorem is the usual replacement for GRH in sieve theory. It controls the remainder terms of a sieve applied to the primes, as in the proof of Chen's theorem. It is also the main input of the Goldston–Pintz–Yıldırım and Maynard methods for bounded gaps between primes. Zhang's proof of bounded gaps needed a variant that goes slightly beyond the level 1/21/2 for smooth moduli. Compare with the Siegel–Walfisz theorem, which is uniform in each single modulus but only for qq up to a power of log⁡x\log x.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Algebraic number field, concept, no evidence levelAlgebraic number fieldconceptBig O notation, concept, no evidence levelBig O notationconceptChebyshev functions, concept, no evidence levelChebyshev functionsconceptDedekind zeta function, concept, no evidence levelDedekind zeta functionconceptDirichlet L-function, concept, no evidence levelDirichlet L-functionconceptDirichlet character, concept, no evidence levelDirichlet characterconceptDivisibility, definition, no evidence levelDivisibilitydefinitionEuler's totient function, concept, no evidence levelEuler's totient funct…conceptLarge sieve, concept, no evidence levelLarge sieveconceptSiegel zero, concept, no evidence levelSiegel zeroconceptZero of a holomorphic function, concept, no evidence levelZero of a holomorphic…conceptvon Mangoldt function, concept, no evidence levelvon Mangoldt functionconceptGeneralized Riemann hypothesis, conjecture, no evidence levelGeneralized Riemann h…conjecturePrime number, definition, no evidence levelPrime numberdefinitionPrime-counting function, definition, no evidence levelPrime-counting functi…definitionPrime number theorem, statement, no evidence levelPrime number theoremstatementSiegel–Walfisz theorem, statement, no evidence levelSiegel–Walfisz theoremstatementBombieri–Vinogradov theorem, statement, no evidence levelBombieri–Vinogradov t…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic number field concept
  2. Big O notation concept
  3. Chebyshev functions concept
  4. Dedekind zeta function concept
  5. Dirichlet L-function concept
  6. Dirichlet character concept
  7. Divisibility definition
  8. Euler's totient function concept
  9. Large sieve concept
  10. Siegel zero concept
  11. Zero of a holomorphic function concept
  12. von Mangoldt function concept
  13. Generalized Riemann hypothesis conjecture
  14. Prime number definition
  15. Prime-counting function definition
  16. Prime number theorem statement
  17. Siegel–Walfisz theorem statement
  18. Bombieri–Vinogradov theorem statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Bombieri–Vinogradov_theorem
About this page
TagB3JMY54GJF
URIm2://B3JMY54GJF
Kindstatement, theorem
WritingW0 stub
MSC11N13, 11N36
StatusThis copy of the site has no build data. Run m2 status B3JMY54GJF.
AboutDirichlet's theorem on arithmetic progressions, Generalized Riemann hypothesis
UsesChebyshev functions, von Mangoldt function, Euler's totient function, Large sieve, Siegel–Walfisz theorem
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Changed2 November 2026

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