Bombieri–Vinogradov theorem
E571012
The Bombieri–Vinogradov theorem is a major result in analytic number theory that gives strong average estimates for the distribution of prime numbers in arithmetic progressions, approaching what is predicted by the Generalized Riemann Hypothesis.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bombieri–Vinogradov theorem canonical | 1 |
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf | theorem in analytic number theory ⓘ |
| appearsIn |
research on distribution of primes in residue classes
ⓘ
research on prime gaps ⓘ |
| appliesTo | moduli q up to about x^{1/2} (with logarithmic savings) ⓘ |
| approximationOf | error term predicted by the Generalized Riemann Hypothesis on average over moduli ⓘ |
| category | results about primes in arithmetic progressions ⓘ |
| comparedWith | Siegel–Walfisz theorem NERFINISHED ⓘ |
| concerns |
distribution of prime numbers in arithmetic progressions
ⓘ
error term in the prime number theorem for arithmetic progressions ⓘ |
| domain | prime numbers ⓘ |
| field | analytic number theory ⓘ |
| generalizationOf | classical average results for primes in arithmetic progressions ⓘ |
| gives | strong average estimates for primes in arithmetic progressions ⓘ |
| hasConsequence |
improved bounds in sieve theory
ⓘ
results on small gaps between primes in arithmetic progressions ⓘ |
| implies | primes are well distributed in arithmetic progressions for most moduli up to about x^{1/2} (up to logarithmic factors) ⓘ |
| improvesOn | Siegel–Walfisz theorem on average over moduli NERFINISHED ⓘ |
| inspired | further work on distribution of primes in arithmetic progressions ⓘ |
| involves |
Dirichlet L-functions
NERFINISHED
ⓘ
Dirichlet characters NERFINISHED ⓘ sums over moduli q ⓘ |
| isToolFor | bounding error terms in arithmetic progression counting functions ⓘ |
| namedAfter |
A. I. Vinogradov
NERFINISHED
ⓘ
Enrico Bombieri NERFINISHED ⓘ |
| provenBy |
A. I. Vinogradov
NERFINISHED
ⓘ
Enrico Bombieri NERFINISHED ⓘ |
| quantifies | average deviation of π(x;q,a) from its expected value x/(φ(q) log x) ⓘ |
| relatedTo |
Dirichlet primes in arithmetic progressions
ⓘ
Elliott–Halberstam conjecture NERFINISHED ⓘ Generalized Riemann Hypothesis NERFINISHED ⓘ |
| requires | zero-free regions for Dirichlet L-functions ⓘ |
| standardReference |
Enrico Bombieri’s paper on the large sieve and its applications to number theory
ⓘ
Iwaniec and Kowalski, Analytic Number Theory NERFINISHED ⓘ |
| status | unconditionally proved theorem ⓘ |
| strengthComparedTo |
comparable to the Generalized Riemann Hypothesis on average over moduli
ⓘ
weaker than the Generalized Riemann Hypothesis pointwise in the modulus ⓘ |
| typeOfResult | average result over moduli ⓘ |
| usedIn |
analytic proofs of results about primes in short intervals
ⓘ
applications to additive problems involving primes ⓘ |
| uses |
large sieve method
ⓘ
zero-density estimates for Dirichlet L-functions ⓘ |
| yearProved | 1965 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.