- Elliott–Halberstam conjecture
In
number theory , the Elliott–Halberstam conjecture is aconjecture about the distribution ofprime number s inarithmetic progression s. It has many applications insieve theory . It is named forPeter D. T. A. Elliott andHeini Halberstam .To state the conjecture requires some notation. Let pi(x) denote the number of primes less than or equal to "x". If "q" is a positive
integer and "a" iscoprime to "q", we let:pi(x;q,a),
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:pi(x;q,a) approx frac{pi(x)}{phi(q)}
when "a" is coprime to "q". If we then define the error function
:E(x;q) = max_{(a,q) = 1} left|pi(x;q,a) - frac{pi(x)}{phi(q)} ight|
where 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 that
:sum_{1 leq q leq x^ heta} E(x;q) leq frac{C x}{log^A x}
for all "x" > 2.
This conjecture was proven for all "θ" < 1/2 by
Enrico Bombieri andA. I. Vinogradov (theBombieri–Vinogradov theorem , sometimes known simply as "Bombieri's theorem"); this result is already quite useful, being an averaged form of thegeneralized 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 recent result of
Dan Goldston , J. Pintz, andCem Yildirim [http://www.arxiv.org/abs/math.NT/0508185] (see also [http://www.arxiv.org/abs/math.NT/0505300] , [http://www.arxiv.org/abs/math.NT/0506067] ), which shows (assuming this conjecture) that there are infinitely many pairs of primes which differ by at most 16.ee also
*
Barban–Davenport–Halberstam theorem
*Barban–Montgomery theorem References
# E. Bombieri, On the large sieve, Mathematika 12 (1965), 201–225
# P.D.T.A. Elliot and H. Halberstam, A conjecture in prime number theory, Symp. Math. 4 (1968-1969), 59-72.
# A.I. Vinogradov, The density hypothesis for Dirichlet L-series (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 903-934.
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