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- Elliott–Halberstam_conjecture abstract "In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam.To state the conjecture requires some notation. Let denote the number of primes less than or equal to x. If q is a positive integer and a is coprime to q, we let , 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 usthat where a is coprime to q and is Euler's totient function. If we then define the error functionwhere the max is taken over all a coprime to q, then the Elliott–Halberstam conjecture is the assertion thatfor every θ < 1 and A > 0 there exists a constant C > 0 such thatfor all x > 2.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.The Elliott–Halberstam conjecture has several consequences. One striking one is the result announced by Dan Goldston, János Pintz, and Cem Yıldırım, 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.".
- Elliott–Halberstam_conjecture wikiPageID "1898427".
- Elliott–Halberstam_conjecture wikiPageRevisionID "583623402".
- Elliott–Halberstam_conjecture subject Category:Analytic_number_theory.
- Elliott–Halberstam_conjecture subject Category:Conjectures_about_prime_numbers.
- Elliott–Halberstam_conjecture comment "In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam.To state the conjecture requires some notation. Let denote the number of primes less than or equal to x. If q is a positive integer and a is coprime to q, we let , denote the number of primes less than or equal to x which are equal to a modulo q.".
- Elliott–Halberstam_conjecture label "Congettura di Elliott-Halberstam".
- Elliott–Halberstam_conjecture label "Conjecture d'Elliott-Halberstam".
- Elliott–Halberstam_conjecture label "Conjetura de Elliott–Halberstam".
- Elliott–Halberstam_conjecture label "Elliott–Halberstam conjecture".
- Elliott–Halberstam_conjecture sameAs Elliott%E2%80%93Halberstam_conjecture.
- Elliott–Halberstam_conjecture sameAs Conjetura_de_Elliott–Halberstam.
- Elliott–Halberstam_conjecture sameAs Conjecture_d'Elliott-Halberstam.
- Elliott–Halberstam_conjecture sameAs Congettura_di_Elliott-Halberstam.
- Elliott–Halberstam_conjecture sameAs Q2993296.
- Elliott–Halberstam_conjecture sameAs Q2993296.
- Elliott–Halberstam_conjecture wasDerivedFrom Elliott–Halberstam_conjecture?oldid=583623402.