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| In [[number theory]], the '''Elliott–Halberstam conjecture''' is a [[conjecture]] about the distribution of [[prime number]]s in [[arithmetic progression]]s. It has many applications in [[sieve theory]]. It is named for [[Peter D. T. A. Elliott]] and [[Heini Halberstam]].
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| To state the conjecture requires some notation. Let <math>\pi(x)</math> denote the number of primes less than or equal to ''x''. If ''q'' is a [[negative and positive numbers|positive]] [[integer]] and ''a'' is [[coprime]] to ''q'', we let <math>\pi(x;q,a)</math>, denote the number of primes less than or equal to ''x'' which are equal to ''a'' modulo ''q''. [[Dirichlet's theorem on primes in arithmetic progressions]] then tells us
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| that
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| :<math> \pi(x;q,a) \approx \frac{\pi(x)}{\varphi(q)}</math>
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| where ''a'' is coprime to ''q'' and <math>\varphi</math> is [[Euler's totient function]]. If we then define the error function
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| :<math> E(x;q) = \max_{(a,q) = 1} \left|\pi(x;q,a) - \frac{\pi(x)}{\varphi(q)}\right|</math>
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| where the max is taken over all ''a'' coprime to ''q'', then the Elliott–Halberstam conjecture is the assertion that
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| for every ''θ'' < 1 and ''A'' > 0 there exists a constant ''C'' > 0 such that
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| :<math> \sum_{1 \leq q \leq x^\theta} E(x;q) \leq \frac{C x}{\log^A x}</math> | |
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| for all ''x'' > 2.
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| This conjecture was proven for all ''θ'' < 1/2 by [[Enrico Bombieri]] and [[A. I. Vinogradov]] (the [[Bombieri–Vinogradov theorem]], sometimes known simply as "Bombieri's theorem"); this result is already quite useful, being an averaged form of the [[generalized Riemann hypothesis]]. It is known that the conjecture fails at the endpoint ''θ'' = 1.
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| The Elliott–Halberstam conjecture has several consequences. One striking one is the result announced by [[Dan Goldston]], [[János Pintz]], and [[Cem Yıldırım]],<ref>{{arxiv|math.NT/0508185}}; see also {{arxiv|math.NT/0505300}}, {{arxiv|math.NT/0506067}}.</ref> which shows (assuming this conjecture) that there are infinitely many pairs of primes which differ by at most 16. In November 2013, James Maynard showed that subject to the Elliott-Halberstam conjecture, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 12.
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| ==See also==
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| *[[Barban–Davenport–Halberstam theorem]]
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| *[[Barban–Montgomery theorem]]
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| ==Notes==
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| {{Reflist}}
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| == References ==
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| *{{cite journal |first=E. |last=Bombieri |title=On the large sieve |journal=Mathematika |volume=12 |issue= |year=1965 |pages=201–225 |doi= }}
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| *{{cite journal |first=P. D. T. A. |last=Elliott |first2=H. |last2=Halberstam |title=A conjecture in prime number theory |journal=Symp. Math. |volume=4 |year=1968 |issue= |pages=59–72 }}
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| *{{cite journal |first=A. I. |last=Vinogradov |title=The density hypothesis for Dirichlet L-series |language=Russian |journal=Izv. Akad. Nauk SSSR Ser. Mat. |volume=29 |issue=4 |year=1965 |pages=903–934 |mr=197414 }}
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| *{{cite journal |first=K. |last=Soundararajan |authorlink=Kannan Soundararajan |title=Small gaps between prime numbers: The work of Goldston–Pintz–Yıldırım |journal=Bull. AMS |volume=44 |year=2007 |issue=1 |pages=1–18 |doi=10.1090/S0273-0979-06-01142-6 }}
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| {{DEFAULTSORT:Elliott-Halberstam conjecture}}
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| [[Category:Analytic number theory]]
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| [[Category:Conjectures about prime numbers]]
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