Lower bound for the remainder in the prime-pair conjecture

2008/06/25 by Jacob Korevaar, Korevaar, Jacob
Mathematics · #11P32 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11P32

paper · pdf · doi:10.48550/arxiv.0806.4057

25 pages, 2 figures

arxiv created 2008/06/25 · arxiv updated 2009/12/01

Abstract

For any positive integer r, let pi_2r(x) denote the number of prime pairs (p, p+2r) with p not exceeding (large) x. According to the prime-pair conjecture of Hardy and Littlewood, pi_2r(x) should be asymptotic to 2C_2rli₂(x) with an explicit positive constant C_2r. A heuristic argument indicates that the remainder e_2r(x) in this approximation cannot be of lower order than x^beta, where beta is the supremum of the real parts of zeta's zeros. The argument also suggests an approximation for pi_2r(x) similar to one of Riemann for pi(x).

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