Rough Numbers Between Consecutive Primes
4 days ago
- #Sieve Methods
- #Prime Gaps
- #Number Theory
- The paper confirms Erdős's prediction that almost all gaps between consecutive primes contain a number with a least prime factor at least the length of the gap.
- The number of exceptional gaps is shown to be at most O(X/log²X), with a more precise asymptotic N(X) ∼ cX/log²X under the Hardy-Littlewood prime tuples conjecture.
- The results rely on sieve-theoretic arguments and asymptotics for singular series developed by Montgomery and Soundararajan.
- The explicit constant c in the asymptotic is believed to be between 2.7 and 2.8.