Author:
Banks William,Ford Kevin,Tao Terence
Abstract
AbstractWe introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval $[1,x]$
[
1
,
x
]
. Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set.
Publisher
Springer Science and Business Media LLC
Reference42 articles.
1. Azuma, K.: Weighted sums of certain dependent random variables. Tohoku Math. J. 19, 357–367 (1967)
2. Baker, R.C., Harman, G., Pintz, J.: The difference between consecutive primes. II. Proc. Lond. Math. Soc. (3) 83(3), 532–562 (2001)
3. Bennett, G.: Probability inequalities for the sum of independent random variables. J. Am. Stat. Assoc. 57, 33–45 (1962)
4. Cadwell, J.H.: Large intervals between consecutive primes. Math. Comput. 25, 909–913 (1971)
5. Cramér, H.: Some theorems concerning prime numbers. Ark. Mat. Astron. Fys. 15(5), 1–32 (1920)