Lower bound for the remainder in the prime-pair conjecture

Details Lower bound for the remainder in the prime-pair conjecture Lower bound for the remainder in the prime-pair conjecture

2008-06-25

arxiv.org/abs/0806.4057v1

Mathematics

Number Theory

25 pages, 2 figures

Scientific paper

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_{2r}li_2(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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